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# Ideals in multivariate polynomial rings.

Sage has a powerful system to compute with multivariate polynomial rings. Most algorithms dealing with these ideals are centered on the computation of Groebner bases. Sage mainly uses Singular to implement this functionality. Singular is widely regarded as the best open-source system for Groebner basis calculation in multivariate polynomial rings over fields.

AUTHORS:

- William Stein
- Kiran S. Kedlaya (2006-02-12): added Macaulay2 analogues of some Singular features
- Martin Albrecht (2008,2007): refactoring, many Singular related functions
- Martin Albrecht (2009): added Groebner basis over rings functionality from Singular 3.1
- John Perry (2012): bug fixing equality & containment of ideals

EXAMPLES:

We compute a Groebner basis for some given ideal. The type returned by the `groebner_basis` method is `PolynomialSequence`, i.e. it is not a `MPolynomialIdeal`:

    sage: x,y,z = QQ['x,y,z'].gens()
    sage: I = ideal(x^5 + y^4 + z^3 - 1,  x^3 + y^3 + z^2 - 1)
    sage: B = I.groebner_basis()
    sage: type(B)
    <class 'sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic'>

Groebner bases can be used to solve the ideal membership problem:

    sage: f,g,h = B
    sage: (2*x*f + g).reduce(B)
    0

    sage: (2*x*f + g) in I
    True

    sage: (2*x*f + 2*z*h + y^3).reduce(B)
    y^3

    sage: (2*x*f + 2*z*h + y^3) in I
    False

We compute a Groebner basis for Cyclic 6, which is a standard benchmark and test ideal.

    sage: R.<x,y,z,t,u,v> = QQ['x,y,z,t,u,v']
    sage: I = sage.rings.ideal.Cyclic(R,6)
    sage: B = I.groebner_basis()
    sage: len(B)
    45

We compute in a quotient of a polynomial ring over \(\ZZ/17\ZZ\):

    sage: R.<x,y> = ZZ[]
    sage: S.<a,b> = R.quotient((x^2 + y^2, 17))
    sage: S
    Quotient of Multivariate Polynomial Ring in x, y over Integer Ring
    by the ideal (x^2 + y^2, 17)

    sage: a^2 + b^2 == 0
    True
    sage: a^3 - b^2
    -a*b^2 - b^2

Note that the result of a computation is not necessarily reduced:

    sage: (a+b)^17
    256*a*b^16 + 256*b^17
    sage: S(17) == 0
    True

Or we can work with \(\ZZ/17\ZZ\) directly:

    sage: R.<x,y> = Zmod(17)[]
    sage: S.<a,b> = R.quotient((x^2 + y^2,))
    sage: S
    Quotient of Multivariate Polynomial Ring in x, y over Ring of
    integers modulo 17 by the ideal (x^2 + y^2)

    sage: a^2 + b^2 == 0
    True
    sage: a^3 - b^2
    -a*b^2 - b^2
    sage: (a+b)^17
    a*b^16 + b^17
    sage: S(17) == 0
    True

Working with a polynomial ring over \(\ZZ\):

    sage: R.<x,y,z,w> = ZZ[]
    sage: I = ideal(x^2 + y^2 - z^2 - w^2, x-y)
    sage: J = I^2
    sage: J.groebner_basis()
    [4*y^4 - 4*y^2*z^2 + z^4 - 4*y^2*w^2 + 2*z^2*w^2 + w^4,
     2*x*y^2 - 2*y^3 - x*z^2 + y*z^2 - x*w^2 + y*w^2,
     x^2 - 2*x*y + y^2]

    sage: y^2 - 2*x*y + x^2 in J
    True
    sage: 0 in J
    True

We do a Groebner basis computation over a number field:

    sage: K.<zeta> = CyclotomicField(3)
    sage: R.<x,y,z> = K[]; R
    Multivariate Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

    sage: i = ideal(x - zeta*y + 1, x^3 - zeta*y^3); i
    Ideal (x + (-zeta)*y + 1, x^3 + (-zeta)*y^3) of Multivariate
    Polynomial Ring in x, y, z over Cyclotomic Field of order 3 and degree 2

    sage: i.groebner_basis()
    [y^3 + (2*zeta + 1)*y^2 + (zeta - 1)*y + (-1/3*zeta - 2/3), x + (-zeta)*y + 1]

    sage: S = R.quotient(i); S
    Quotient of Multivariate Polynomial Ring in x, y, z over
    Cyclotomic Field of order 3 and degree 2 by the ideal (x +
    (-zeta)*y + 1, x^3 + (-zeta)*y^3)

    sage: S.0  - zeta*S.1
    -1
    sage: S.0^3 - zeta*S.1^3
    0

Two examples from the Mathematica documentation (done in Sage):

> We compute a Groebner basis:

>     sage: R.<x,y> = PolynomialRing(QQ, order='lex')
>     sage: ideal(x^2 - 2*y^2, x*y - 3).groebner_basis()
>     [x - 2/3*y^3, y^4 - 9/2]

> We show that three polynomials have no common root:

>     sage: R.<x,y> = QQ[]
>     sage: ideal(x+y, x^2 - 1, y^2 - 2*x).groebner_basis()
>     [1]

The next example shows how we can use Groebner bases over \(\ZZ\) to find the primes modulo which a system of equations has a solution, when the system has no solutions over the rationals.

> We first form a certain ideal \(I\) in \(\ZZ[x, y, z]\), and note that the Groebner basis of \(I\) over \(\QQ\) contains 1, so there are no solutions over \(\QQ\) or an algebraic closure of it (this is not surprising as there are 4 equations in 3 unknowns).

>     sage: P.<x,y,z> = PolynomialRing(ZZ,order='lex')
>     sage: I = ideal(-y^2 - 3*y + z^2 + 3, -2*y*z + z^2 + 2*z + 1, \
>                     x*z + y*z + z^2, -3*x*y + 2*y*z + 6*z^2)
>     sage: I.change_ring(P.change_ring(QQ)).groebner_basis()
>     [1]

> However, when we compute the Groebner basis of I (defined over \(\ZZ\)), we note that there is a certain integer in the ideal which is not 1.

>     sage: I.groebner_basis()
>     [x + 130433*y + 59079*z, y^2 + 3*y + 17220, y*z + 5*y + 14504, 2*y + 158864, z^2 + 17223, 2*z + 41856, 164878]

> Now for each prime \(p\) dividing this integer 164878, the Groebner basis of I modulo \(p\) will be non-trivial and will thus give a solution of the original system modulo \(p\).

>     sage: factor(164878)
>     2 * 7 * 11777

>     sage: I.change_ring(P.change_ring( GF(2) )).groebner_basis()
>     [x + y + z, y^2 + y, y*z + y, z^2 + 1]
>     sage: I.change_ring(P.change_ring( GF(7) )).groebner_basis()
>     [x - 1, y + 3, z - 2]
>     sage: I.change_ring(P.change_ring( GF(11777 ))).groebner_basis()
>     [x + 5633, y - 3007, z - 2626]

> The Groebner basis modulo any product of the prime factors is also non-trivial.

>     sage: I.change_ring(P.change_ring( IntegerModRing(2*7) )).groebner_basis()
>     [x + y + z, y^2 + 3*y, y*z + 11*y + 4, 2*y + 6, z^2 + 3, 2*z + 10]

> Modulo any other prime the Groebner basis is trivial so there are no other solutions. For example:

>     sage: I.change_ring( P.change_ring( GF(3) ) ).groebner_basis()
>     [1]

TESTS:

    sage: x,y,z = QQ['x,y,z'].gens()
    sage: I = ideal(x^5 + y^4 + z^3 - 1,  x^3 + y^3 + z^2 - 1)
    sage: I == loads(dumps(I))
    True

Note

Sage distinguishes between lists or sequences of polynomials and ideals. Thus an ideal is not identified with a particular set of generators. For sequences of multivariate polynomials see `sage.rings.polynomial.multi_polynomial_sequence.PolynomialSequence_generic`.

class `sage.rings.polynomial.multi_polynomial_ideal.``LibSingularDefaultContext`(singular=Singular)

class `sage.rings.polynomial.multi_polynomial_ideal.``MPolynomialIdeal`(ring, gens, coerce=True)

  Bases: `sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr`, `sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_macaulay2_repr`, `sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_magma_repr`, `sage.rings.ideal.Ideal_generic`

  Create an ideal in a multivariate polynomial ring.

  INPUT:

  - `ring` - the ring the ideal is defined in
  - `gens` - a list of generators for the ideal
  - `coerce` - coerce elements to the ring `ring`?

  EXAMPLES:

      sage: R.<x,y> = PolynomialRing(IntegerRing(), 2, order='lex')
      sage: R.ideal([x, y])
      Ideal (x, y) of Multivariate Polynomial Ring in x, y over Integer Ring
      sage: R.<x0,x1> = GF(3)[]
      sage: R.ideal([x0^2, x1^3])
      Ideal (x0^2, x1^3) of Multivariate Polynomial Ring in x0, x1 over Finite Field of size 3

  `change_ring`(P)

    Return the ideal `I` in `P` spanned by the generators \(g_1, ..., g_n\) of self as returned by `self.gens()`.

    INPUT:

    - `P` - a multivariate polynomial ring

    EXAMPLE:

        sage: P.<x,y,z> = PolynomialRing(QQ,3,order='lex')
        sage: I = sage.rings.ideal.Cyclic(P)
        sage: I
        Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
        Multivariate Polynomial Ring in x, y, z over Rational Field

        sage: I.groebner_basis()
        [x + y + z, y^2 + y*z + z^2, z^3 - 1]

        sage: Q.<x,y,z> = P.change_ring(order='degrevlex'); Q
        Multivariate Polynomial Ring in x, y, z over Rational Field
        sage: Q.term_order()
        Degree reverse lexicographic term order

        sage: J = I.change_ring(Q); J
        Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1) of
        Multivariate Polynomial Ring in x, y, z over Rational Field

        sage: J.groebner_basis()
        [z^3 - 1, y^2 + y*z + z^2, x + y + z]

  `degree_of_semi_regularity`()

    Return the degree of semi-regularity of this ideal under the assumption that it is semi-regular.

    Let \(\{f_1, ... , f_m\} \subset K[x_1 , ... , x_n]\) be homogeneous polynomials of degrees \(d_1,... ,d_m\) respectively. This sequence is semi-regular if:

    > - \(\{f_1, ... , f_m\} \neq K[x_1 , ... , x_n]\)
    > - for all \(1 \leq i \leq m\) and \(g \in K[x_1,\dots,x_n]\): \(deg(g \cdot pi ) < D\) and \(g \cdot f_i \in <f_1 , \dots , f_{i-1}>\) implies that \(g \in <f_1, ..., f_{i-1}>\) where \(D\) is the degree of regularity.

    This notion can be extended to affine polynomials by considering their homogeneous components of highest degree.

    The degree of regularity of a semi-regular sequence \(f_1, ...,f_m\) of respective degrees \(d_1,...,d_m\) is given by the index of the first non-positive coefficient of:

    > \(\sum c_k z^k = \frac{\prod (1 - z^{d_i})}{(1-z)^n}\)

    EXAMPLE:

    We consider a homogeneous example:

        sage: n = 8
        sage: K = GF(127)
        sage: P = PolynomialRing(K,n,'x')
        sage: s = [K.random_element() for _ in range(n)]
        sage: L = []
        sage: for i in range(2*n):
        ....:     f = P.random_element(degree=2, terms=binomial(n,2))
        ....:     f -= f(*s)
        ....:     L.append(f.homogenize())
        sage: I = Ideal(L)
        sage: I.degree_of_semi_regularity()
        4

    From this, we expect a Groebner basis computation to reach at most degree 4. For homogeneous systems this is equivalent to the largest degree in the Groebner basis:

        sage: max(f.degree() for f in I.groebner_basis())
        4

    We increase the number of polynomials and observe a decrease the degree of regularity:

        sage: for i in range(2*n):
        ....:     f = P.random_element(degree=2, terms=binomial(n,2))
        ....:     f -= f(*s)
        ....:     L.append(f.homogenize())
        sage: I = Ideal(L)
        sage: I.degree_of_semi_regularity()
        3

        sage: max(f.degree() for f in I.groebner_basis())
        3

    The degree of regularity approaches 2 for quadratic systems as the number of polynomials approaches \(n^2\):

        sage: for i in range((n-4)*n):
        ....:     f = P.random_element(degree=2, terms=binomial(n,2))
        ....:     f -= f(*s)
        ....:     L.append(f.homogenize())
        sage: I = Ideal(L)
        sage: I.degree_of_semi_regularity()
        2

        sage: max(f.degree() for f in I.groebner_basis())
        2

    Note

    It is unknown whether semi-regular sequences exist. However, it is expected that random systems are semi-regular sequences. For more details about semi-regular sequences see [BFS04].

    REFERENCES:

    [BFS04] | Magali Bardet, Jean-Charles Faugère, and Bruno Salvy, On the complexity of Groebner basis computation of semi-regular overdetermined algebraic equations. Proc. International Conference on Polynomial System Solving (ICPSS), pp. 71-75, 2004.

  `gens`()

    Return a set of generators / a basis of self. This is usually the set of generators provided during object creation.

    EXAMPLE:

        sage: P.<x,y> = PolynomialRing(QQ,2)
        sage: I = Ideal([x,y+1]); I
        Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
        sage: I.gens()
        [x, y + 1]

  `groebner_basis`(algorithm='', deg_bound=None, mult_bound=None, prot=False, *args, **kwds)

    Return the reduced Groebner basis of this ideal.

    A Groebner basis \(g_1,...,g_n\) for an ideal \(I\) is a generating set such that \(<LM(g_i)> = LM(I)\), i.e., the leading monomial ideal of \(I\) is spanned by the leading terms of \(g_1,...,g_n\). Groebner bases are the key concept in computational ideal theory in multivariate polynomial rings which allows a variety of problems to be solved.

    Additionally, a reduced Groebner basis \(G\) is a unique representation for the ideal \(<G>\) with respect to the chosen monomial ordering.

    INPUT:

    - `algorithm` - determines the algorithm to use, see below

        for available algorithms.

    - `deg_bound` - only compute to degree `deg_bound`, that is, ignore all S-polynomials of higher degree. (default: `None`)

    - `mult_bound` - the computation is stopped if the ideal is zero-dimensional in a ring with local ordering and its multiplicity is lower than `mult_bound`. Singular only. (default: `None`)

    - `prot` - if set to `True` the computation protocol of the underlying implementation is printed. If an algorithm from the `singular:` or `magma:` family is used, `prot` may also be `sage` in which case the output is parsed and printed in a common format where the amount of information printed can be controlled via calls to `set_verbose()`.

    - `*args` - additional parameters passed to the respective

        implementations

    - `**kwds` - additional keyword parameters passed to the

        respective implementations

    ALGORITHMS:

    ‘’
      autoselect (default)
    ‘singular:groebner’
      Singular’s `groebner` command
    ‘singular:std’
      Singular’s `std` command
    ‘singular:stdhilb’
      Singular’s `stdhib` command
    ‘singular:stdfglm’
      Singular’s `stdfglm` command
    ‘singular:slimgb’
      Singular’s `slimgb` command
    ‘libsingular:groebner’
      libSingular’s `groebner` command
    ‘libsingular:std’
      libSingular’s `std` command
    ‘libsingular:slimgb’
      libSingular’s `slimgb` command
    ‘libsingular:stdhilb’
      libSingular’s `stdhib` command
    ‘libsingular:stdfglm’
      libSingular’s `stdfglm` command
    ‘toy:buchberger’
      Sage’s toy/educational buchberger without Buchberger criteria
    ‘toy:buchberger2’
      Sage’s toy/educational buchberger with Buchberger criteria
    ‘toy:d_basis’
      Sage’s toy/educational algorithm for computation over PIDs
    ‘macaulay2:gb’
      Macaulay2’s `gb` command (if available)
    ‘magma:GroebnerBasis’
      Magma’s `Groebnerbasis` command (if available)
    ‘ginv:TQ’, ‘ginv:TQBlockHigh’, ‘ginv:TQBlockLow’ and ‘ginv:TQDegree’
      One of GINV’s implementations (if available)

    If only a system is given - e.g. ‘magma’ - the default algorithm is chosen for that system.

    Note

    The Singular and libSingular versions of the respective algorithms are identical, but the former calls an external Singular process while the later calls a C function, i.e. the calling overhead is smaller. However, the libSingular interface does not support pretty printing of computation protocols.

    EXAMPLES:

    Consider Katsura-3 over \(\QQ\) with lexicographical term ordering. We compute the reduced Groebner basis using every available implementation and check their equality.

        sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex')
        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis()
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('libsingular:groebner')
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('libsingular:std')
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('libsingular:stdhilb')
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('libsingular:stdfglm')
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('libsingular:slimgb')
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

    Note that `toy:buchberger` does not return the reduced Groebner basis,

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('toy:buchberger')
        [a^2 - a + 2*b^2 + 2*c^2,
         a*b + b*c - 1/2*b, a + 2*b + 2*c - 1,
         b^2 + 3*b*c - 1/2*b + 3*c^2 - c,
         b*c - 1/10*b + 6/5*c^2 - 2/5*c,
         b + 30*c^3 - 79/7*c^2 + 3/7*c,
         c^6 - 79/210*c^5 - 229/2100*c^4 + 121/2520*c^3 + 1/3150*c^2 - 11/12600*c,
         c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

    but that `toy:buchberger2` does.:

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('toy:buchberger2')
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('macaulay2:gb') # optional - macaulay2
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

        sage: I = sage.rings.ideal.Katsura(P,3) # regenerate to prevent caching
        sage: I.groebner_basis('magma:GroebnerBasis') # optional - magma
        [a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]

    Singular and libSingular can compute Groebner basis with degree restrictions.:

        sage: R.<x,y> = QQ[]
        sage: I = R*[x^3+y^2,x^2*y+1]
        sage: I.groebner_basis(algorithm='singular')
        [x^3 + y^2, x^2*y + 1, y^3 - x]
        sage: I.groebner_basis(algorithm='singular',deg_bound=2)
        [x^3 + y^2, x^2*y + 1]
        sage: I.groebner_basis()
        [x^3 + y^2, x^2*y + 1, y^3 - x]
        sage: I.groebner_basis(deg_bound=2)
        [x^3 + y^2, x^2*y + 1]

    A protocol is printed, if the verbosity level is at least 2, or if the argument `prot` is provided. Historically, the protocol did not appear during doctests, so, we skip the examples with protocol output.

        sage: set_verbose(2)
        sage: I = R*[x^3+y^2,x^2*y+1]
        sage: I.groebner_basis()  # not tested
        std in (0),(x,y),(dp(2),C)
        [...:2]3ss4s6
        (S:2)--
        product criterion:1 chain criterion:0
        [x^3 + y^2, x^2*y + 1, y^3 - x]
        sage: I.groebner_basis(prot=False)
        std in (0),(x,y),(dp(2),C)
        [...:2]3ss4s6
        (S:2)--
        product criterion:1 chain criterion:0
        [x^3 + y^2, x^2*y + 1, y^3 - x]
        sage: set_verbose(0)
        sage: I.groebner_basis(prot=True)  # not tested
        std in (0),(x,y),(dp(2),C)
        [...:2]3ss4s6
        (S:2)--
        product criterion:1 chain criterion:0
        [x^3 + y^2, x^2*y + 1, y^3 - x]

    The list of available options is provided at `LibSingularOptions`.

    Note that Groebner bases over \(\ZZ\) can also be computed.:

        sage: P.<a,b,c> = PolynomialRing(ZZ,3)
        sage: I = P * (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b)
        sage: I.groebner_basis()
        [b^3 - 23*b*c^2 + 3*b^2 + 5*b*c, 2*b*c^2 - 6*c^3 - b^2 - b*c + 2*c^2,
         42*c^3 + 5*b^2 + 4*b*c - 14*c^2, 2*b^2 + 6*b*c + 6*c^2 - b - 2*c,
         10*b*c + 12*c^2 - b - 4*c, a + 2*b + 2*c - 1]

        sage: I.groebner_basis('macaulay2') # optional - macaulay2
        [b^3 + b*c^2 + 12*c^3 + b^2 + b*c - 4*c^2,
         2*b*c^2 - 6*c^3 + b^2 + 5*b*c + 8*c^2 - b - 2*c,
         42*c^3 + b^2 + 2*b*c - 14*c^2 + b,
         2*b^2 - 4*b*c - 6*c^2 + 2*c, 10*b*c + 12*c^2 - b - 4*c,
         a + 2*b + 2*c - 1]

    Groebner bases over \(\ZZ/n\ZZ\) are also supported:

        sage: P.<a,b,c> = PolynomialRing(Zmod(1000),3)
        sage: I = P * (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b)
        sage: I.groebner_basis()
        [b*c^2 + 992*b*c + 712*c^2 + 332*b + 96*c,
         2*c^3 + 589*b*c + 862*c^2 + 762*b + 268*c,
         b^2 + 438*b*c + 281*b,
         5*b*c + 156*c^2 + 112*b + 948*c,
         50*c^2 + 600*b + 650*c, a + 2*b + 2*c + 999, 125*b]

    Sage also supports local orderings:

        sage: P.<x,y,z> = PolynomialRing(QQ,3,order='negdegrevlex')
        sage: I = P * (  x*y*z + z^5, 2*x^2 + y^3 + z^7, 3*z^5 +y ^5 )
        sage: I.groebner_basis()
        [x^2 + 1/2*y^3, x*y*z + z^5, y^5 + 3*z^5, y^4*z - 2*x*z^5, z^6]

    We can represent every element in the ideal as a combination of the generators using the `lift()` method:

        sage: P.<x,y,z> = PolynomialRing(QQ,3)
        sage: I = P * ( x*y*z + z^5, 2*x^2 + y^3 + z^7, 3*z^5 +y ^5 )
        sage: J = Ideal(I.groebner_basis())
        sage: f = sum(P.random_element(terms=2)*f for f in I.gens())
        sage: f
        1/2*y^2*z^7 - 1/4*y*z^8 + 2*x*z^5 + 95*z^6 + 1/2*y^5 - 1/4*y^4*z + x^2*y^2 + 3/2*x^2*y*z + 95*x*y*z^2
        sage: f.lift(I.gens())
        [2*x + 95*z, 1/2*y^2 - 1/4*y*z, 0]
        sage: l = f.lift(J.gens()); l
        [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1/2*y^2 + 1/4*y*z, 1/2*y^2*z^2 - 1/4*y*z^3 + 2*x + 95*z]
        sage: sum(map(mul, zip(l,J.gens()))) == f
        True

    Groebner bases over fraction fields of polynomial rings are also supported:

        sage: P.<t> = QQ[]
        sage: F = Frac(P)
        sage: R.<X,Y,Z> = F[]
        sage: I = Ideal([f + P.random_element() for f in sage.rings.ideal.Katsura(R).gens()])
        sage: I.groebner_basis()
        [Z^3 + (79/105*t^2 - 79/105*t + 79/630)*Z^2 + (-11/105*t^4 + 22/105*t^3 - 17/45*t^2 + 197/630*t + 557/1890)*Y + ...,
        Y^2 + (-3/5)*Z^2 + (2/5*t^2 - 2/5*t + 1/15)*Y + (-2/5*t^2 + 2/5*t - 1/15)*Z - 1/10*t^4 + 1/5*t^3 - 7/30*t^2 + 2/5*t + 11/90,
        Y*Z + 6/5*Z^2 + (1/5*t^2 - 1/5*t + 1/30)*Y + (4/5*t^2 - 4/5*t + 2/15)*Z + 1/5*t^4 - 2/5*t^3 + 7/15*t^2 - 3/10*t - 11/45, X + 2*Y + 2*Z + t^2 - t - 1/3]

    In cases where a characteristic cannot be determined, we use a toy implementation of Buchberger’s algorithm (see trac ticket #6581):

        sage: R.<a,b> = QQ[]; I = R.ideal(a^2+b^2-1)
        sage: Q = QuotientRing(R,I); K = Frac(Q)
        sage: R2.<x,y> = K[]; J = R2.ideal([(a^2+b^2)*x + y, x+y])
        sage: J.groebner_basis()
        verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
        [x + y]

    ALGORITHM:

    Uses Singular, Magma (if available), Macaulay2 (if available), or a toy implementation.

  `groebner_fan`(is_groebner_basis=False, symmetry=None, verbose=False)

    Return the Groebner fan of this ideal.

    The base ring must be \(\QQ\) or a finite field \(\GF{p}\) of with \(p \leq 32749\).

    EXAMPLES:

        sage: P.<x,y> = PolynomialRing(QQ)
        sage: i = ideal(x^2 - y^2 + 1)
        sage: g = i.groebner_fan()
        sage: g.reduced_groebner_bases()
        [[x^2 - y^2 + 1], [-x^2 + y^2 - 1]]

    INPUT:

    - `is_groebner_basis` - bool (default False). if True, then I.gens() must be a Groebner basis with respect to the standard degree lexicographic term order.
    - `symmetry` - default: None; if not None, describes symmetries of the ideal
    - `verbose` - default: False; if True, printout useful info during computations
  `homogenize`(var='h')

    Return homogeneous ideal spanned by the homogeneous polynomials generated by homogenizing the generators of this ideal.

    INPUT:

    - `h` - variable name or variable in cover ring (default: ‘h’)

    EXAMPLE:

        sage: P.<x,y,z> = PolynomialRing(GF(2))
        sage: I = Ideal([x^2*y + z + 1, x + y^2 + 1]); I
        Ideal (x^2*y + z + 1, y^2 + x + 1) of Multivariate
        Polynomial Ring in x, y, z over Finite Field of size 2

        sage: I.homogenize()
        Ideal (x^2*y + z*h^2 + h^3, y^2 + x*h + h^2) of
        Multivariate Polynomial Ring in x, y, z, h over Finite
        Field of size 2

        sage: I.homogenize(y)
        Ideal (x^2*y + y^3 + y^2*z, x*y) of Multivariate
        Polynomial Ring in x, y, z over Finite Field of size 2

               sage: I = Ideal([x^2*y + z^3 + y^2*x, x + y^2 + 1])
        sage: I.homogenize()
        Ideal (x^2*y + x*y^2 + z^3, y^2 + x*h + h^2) of
        Multivariate Polynomial Ring in x, y, z, h over Finite
        Field of size 2

  `is_homogeneous`()

    Return `True` if this ideal is spanned by homogeneous polynomials, i.e. if it is a homogeneous ideal.

    EXAMPLE:

        sage: P.<x,y,z> = PolynomialRing(QQ,3)
        sage: I = sage.rings.ideal.Katsura(P)
        sage: I
        Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y +
        2*y*z - y) of Multivariate Polynomial Ring in x, y, z over
        Rational Field

        sage: I.is_homogeneous()
        False

        sage: J = I.homogenize()
        sage: J
        Ideal (x + 2*y + 2*z - h, x^2 + 2*y^2 + 2*z^2 - x*h, 2*x*y
        + 2*y*z - y*h) of Multivariate Polynomial Ring in x, y, z,
        h over Rational Field

        sage: J.is_homogeneous()
        True

  `plot`(*args, **kwds)

    Plot the real zero locus of this principal ideal.

    INPUT:

    - `self` - a principal ideal in 2 variables

    - `algorithm` - set this to ‘surf’ if you want ‘surf’ to

        plot the ideal (default: None)

    - `*args` - optional tuples `(variable, minimum, maximum)`

        for plotting dimensions

    - `**kwds` - optional keyword arguments passed on to

        `implicit_plot`

    EXAMPLES:

    Implicit plotting in 2-d:

        sage: R.<x,y> = PolynomialRing(QQ,2)
        sage: I = R.ideal([y^3 - x^2])
        sage: I.plot()                         # cusp
        Graphics object consisting of 1 graphics primitive

        sage: I = R.ideal([y^2 - x^2 - 1])
        sage: I.plot((x,-3, 3), (y, -2, 2))  # hyperbola
        Graphics object consisting of 1 graphics primitive

        sage: I = R.ideal([y^2 + x^2*(1/4) - 1])
        sage: I.plot()                         # ellipse
        Graphics object consisting of 1 graphics primitive

        sage: I = R.ideal([y^2-(x^2-1)*(x-2)])
        sage: I.plot()                         # elliptic curve
        Graphics object consisting of 1 graphics primitive

        sage: f = ((x+3)^3 + 2*(x+3)^2 - y^2)*(x^3 - y^2)*((x-3)^3-2*(x-3)^2-y^2)
        sage: I = R.ideal(f)
        sage: I.plot()                         # the Singular logo
        Graphics object consisting of 1 graphics primitive

    This used to be trac ticket #5267:

        sage: I = R.ideal([-x^2*y+1])
        sage: I.plot()
        Graphics object consisting of 1 graphics primitive

    AUTHORS:

    - Martin Albrecht (2008-09)
  `reduce`(f)

    Reduce an element modulo the reduced Groebner basis for this ideal. This returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

    EXAMPLES:

        sage: R.<x,y> = PolynomialRing(QQ, 2)
        sage: I = (x^3 + y, y)*R
        sage: I.reduce(y)
        0
        sage: I.reduce(x^3)
        0
        sage: I.reduce(x - y)
        x

        sage: I = (y^2 - (x^3 + x))*R
        sage: I.reduce(x^3)
        y^2 - x
        sage: I.reduce(x^6)
        y^4 - 2*x*y^2 + x^2
        sage: (y^2 - x)^2
        y^4 - 2*x*y^2 + x^2

    Note

    Requires computation of a Groebner basis, which can be a very expensive operation.

  `subs`(in_dict=None, **kwds)

    Substitute variables.

    This method substitutes some variables in the polynomials that generate the ideal with given values. Variables that are not specified in the input remain unchanged.

    INPUT:

    - `in_dict` – (optional) dictionary of inputs
    - `**kwds` – named parameters

    OUTPUT:

    A new ideal with modified generators. If possible, in the same polynomial ring. Raises a `TypeError` if no common polynomial ring of the substituted generators can be found.

    EXAMPLES:

        sage: R.<x,y> = PolynomialRing(ZZ,2,'xy')
        sage: I = R.ideal(x^5+y^5, x^2 + y + x^2*y^2 + 5); I
        Ideal (x^5 + y^5, x^2*y^2 + x^2 + y + 5) of Multivariate Polynomial Ring in x, y over Integer Ring
        sage: I.subs(x=y)
        Ideal (2*y^5, y^4 + y^2 + y + 5) of Multivariate Polynomial Ring in x, y over Integer Ring
        sage: I.subs({x:y})    # same substitution but with dictionary
        Ideal (2*y^5, y^4 + y^2 + y + 5) of Multivariate Polynomial Ring in x, y over Integer Ring

    The new ideal can be in a different ring:

        sage: R.<a,b> = PolynomialRing(QQ,2)
        sage: S.<x,y> = PolynomialRing(QQ,2)
        sage: I = R.ideal(a^2+b^2+a-b+2); I
        Ideal (a^2 + b^2 + a - b + 2) of Multivariate Polynomial Ring in a, b over Rational Field
        sage: I.subs(a=x, b=y)
        Ideal (x^2 + y^2 + x - y + 2) of Multivariate Polynomial Ring in x, y over Rational Field

    The resulting ring need not be a mulitvariate polynomial ring:

        sage: T.<t> = PolynomialRing(QQ)
        sage: I.subs(a=t, b=t)
        Principal ideal (t^2 + 1) of Univariate Polynomial Ring in t over Rational Field
        sage: var("z")
        z
        sage: I.subs(a=z, b=z)
        Principal ideal (2*z^2 + 2) of Symbolic Ring

    Variables that are not substituted remain unchanged:

        sage: R.<x,y> = PolynomialRing(QQ,2)
        sage: I = R.ideal(x^2+y^2+x-y+2); I
        Ideal (x^2 + y^2 + x - y + 2) of Multivariate Polynomial Ring in x, y over Rational Field
        sage: I.subs(x=1)
        Ideal (y^2 - y + 4) of Multivariate Polynomial Ring in x, y over Rational Field

  `weil_restriction`()

    Compute the Weil restriction of this ideal over some extension field.

    A Weil restriction of scalars - denoted \(Res_{L/k}\) - is a functor which, for any finite extension of fields \(L/k\) and any algebraic variety \(X\) over \(L\), produces another corresponding variety \(Res_{L/k}(X)\), defined over \(k\). It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields.

    This function does not compute this Weil restriction directly but computes on generating sets of polynomial ideals:

    Let \(d\) be the degree of the field extension \(L/k\), let \(a\) a generator of \(L/k\) and \(p\) the minimal polynomial of \(L/k\). Denote this ideal by \(I\).

    Specifically, this function first maps each variable \(x\) to its representation over \(k\): \(\sum_{i=0}^{d-1} a^i x_i\). Then each generator of \(I\) is evaluated over these representations and reduced modulo the minimal polynomial \(p\). The result is interpreted as a univariate polynomial in \(a\) and its coefficients are the new generators of the returned ideal.

    If the input and the output ideals are radical, this is equivalent to the statement about algebraic varieties above.

    EXAMPLE:

        sage: k.<a> = GF(2^2)
        sage: P.<x,y> = PolynomialRing(k,2)
        sage: I = Ideal([x*y + 1, a*x + 1])
        sage: I.variety()
        [{y: a, x: a + 1}]
        sage: J = I.weil_restriction()
        sage: J
        Ideal (x1*y0 + x0*y1 + x1*y1, x0*y0 + x1*y1 + 1, x0 + x1, x1 + 1) of
        Multivariate Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2
        sage: J += sage.rings.ideal.FieldIdeal(J.ring()) # ensure radical ideal
        sage: J.variety()
        [{y1: 1, x1: 1, x0: 1, y0: 0}]

        sage: J.weil_restriction() # returns J
        Ideal (x1*y0 + x0*y1 + x1*y1, x0*y0 + x1*y1 + 1, x0 + x1, x1 + 1, x0^2 + x0,
               x1^2 + x1, y0^2 + y0, y1^2 + y1)
               of Multivariate Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2

        sage: k.<a> = GF(3^5)
        sage: P.<x,y,z> = PolynomialRing(k)
        sage: I = sage.rings.ideal.Katsura(P)
        sage: I.dimension()
        0
        sage: I.variety()
        [{y: 0, z: 0, x: 1}]

        sage: J = I.weil_restriction(); J
        Ideal (x4 - y4 - z4, x3 - y3 - z3, x2 - y2 - z2, x1 - y1 - z1, x0 - y0 - z0 - 1,
               x2^2 - x1*x3 - x0*x4 + x4^2 - y2^2 + y1*y3 + y0*y4 - y4^2 - z2^2 + z1*z3 + z0*z4 - z4^2 - x4,
               -x1*x2 - x0*x3 - x3*x4 - x4^2 + y1*y2 + y0*y3 + y3*y4 + y4^2 + z1*z2 + z0*z3 + z3*z4 + z4^2 - x3,
               x1^2 - x0*x2 + x3^2 - x2*x4 + x3*x4 - y1^2 + y0*y2 - y3^2 + y2*y4 - y3*y4 - z1^2 + z0*z2 - z3^2 + z2*z4 - z3*z4 - x2,
               -x0*x1 - x2*x3 - x3^2 - x1*x4 + x2*x4 + y0*y1 + y2*y3 + y3^2 + y1*y4 - y2*y4 + z0*z1 + z2*z3 + z3^2 + z1*z4 - z2*z4 - x1,
               x0^2 + x2*x3 + x1*x4 - y0^2 - y2*y3 - y1*y4 - z0^2 - z2*z3 - z1*z4 - x0,
               -x4*y0 - x3*y1 - x2*y2 - x1*y3 - x0*y4 - x4*y4 - y4*z0 - y3*z1 - y2*z2 - y1*z3 - y0*z4 - y4*z4 - y4,
               -x3*y0 - x2*y1 - x1*y2 - x0*y3 - x4*y3 - x3*y4 + x4*y4 - y3*z0 - y2*z1 - y1*z2 - y0*z3 - y4*z3 - y3*z4 + y4*z4 - y3,
               -x2*y0 - x1*y1 - x0*y2 - x4*y2 - x3*y3 + x4*y3 - x2*y4 + x3*y4 - y2*z0 - y1*z1 - y0*z2 - y4*z2 - y3*z3 + y4*z3 - y2*z4 + y3*z4 - y2,
               -x1*y0 - x0*y1 - x4*y1 - x3*y2 + x4*y2 - x2*y3 + x3*y3 - x1*y4 + x2*y4 - y1*z0 - y0*z1 - y4*z1 - y3*z2 + y4*z2 - y2*z3 + y3*z3 - y1*z4 + y2*z4 - y1,
               -x0*y0 + x4*y1 + x3*y2 + x2*y3 + x1*y4 - y0*z0 + y4*z1 + y3*z2 + y2*z3 + y1*z4 - y0) of Multivariate Polynomial Ring in
               x0, x1, x2, x3, x4, y0, y1, y2, y3, y4, z0, z1, z2, z3, z4 over Finite Field of size 3
        sage: J += sage.rings.ideal.FieldIdeal(J.ring()) # ensure radical ideal
        sage: J.variety()
        [{y1: 0, y4: 0, x4: 0, y2: 0, y3: 0, y0: 0, x2: 0, z4: 0, z3: 0, z2: 0, x1: 0, z1: 0, z0: 0, x0: 1, x3: 0}]

    Weil restrictions are often used to study elliptic curves over extension fields so we give a simple example involving those:

        sage: K.<a> = QuadraticField(1/3)
        sage: E = EllipticCurve(K,[1,2,3,4,5])

    We pick a point on `E`:

        sage: p = E.lift_x(1); p
        (1 : 2 : 1)

        sage: I = E.defining_ideal(); I
        Ideal (-x^3 - 2*x^2*z + x*y*z + y^2*z - 4*x*z^2 + 3*y*z^2 - 5*z^3)
        of Multivariate Polynomial Ring in x, y, z over Number Field in a with defining polynomial x^2 - 1/3

    Of course, the point `p` is a root of all generators of `I`:

        sage: I.subs(x=1,y=2,z=1)
        Ideal (0) of Multivariate Polynomial Ring in x, y, z over
        Number Field in a with defining polynomial x^2 - 1/3

    `I` is also radical:

        sage: I.radical() == I
        True

    So we compute its Weil restriction:

        sage: J = I.weil_restriction()
        sage: J
        Ideal (-3*x0^2*x1 - 1/3*x1^3 - 4*x0*x1*z0 + x1*y0*z0 + x0*y1*z0 + 2*y0*y1*z0 - 4*x1*z0^2 + 3*y1*z0^2 - 2*x0^2*z1
                - 2/3*x1^2*z1 + x0*y0*z1 + y0^2*z1 + 1/3*x1*y1*z1 + 1/3*y1^2*z1 - 8*x0*z0*z1 + 6*y0*z0*z1 - 15*z0^2*z1 - 4/3*x1*z1^2 + y1*z1^2 - 5/3*z1^3,
               -x0^3 - x0*x1^2 - 2*x0^2*z0 - 2/3*x1^2*z0 + x0*y0*z0 + y0^2*z0 + 1/3*x1*y1*z0 + 1/3*y1^2*z0 - 4*x0*z0^2 + 3*y0*z0^2 - 5*z0^3 - 4/3*x0*x1*z1
                + 1/3*x1*y0*z1 + 1/3*x0*y1*z1 + 2/3*y0*y1*z1 - 8/3*x1*z0*z1 + 2*y1*z0*z1 - 4/3*x0*z1^2 + y0*z1^2 - 5*z0*z1^2)
        of Multivariate Polynomial Ring in x0, x1, y0, y1, z0, z1 over Rational Field

    We can check that the point `p` is still a root of all generators of `J`:

        sage: J.subs(x0=1,y0=2,z0=1,x1=0,y1=0,z1=0)
        Ideal (0, 0) of Multivariate Polynomial Ring in x0, x1, y0, y1, z0, z1 over Rational Field

    Note

    Based on a Singular implementation by Michael Brickenstein

class `sage.rings.polynomial.multi_polynomial_ideal.``MPolynomialIdeal_macaulay2_repr`

  An ideal in a multivariate polynomial ring, which has an underlying Macaulay2 ring associated to it.

  EXAMPLES:

      sage: R.<x,y,z,w> = PolynomialRing(ZZ, 4)
      sage: I = ideal(x*y-z^2, y^2-w^2)
      sage: I
      Ideal (x*y - z^2, y^2 - w^2) of Multivariate Polynomial Ring in x, y, z, w over Integer Ring

class `sage.rings.polynomial.multi_polynomial_ideal.``MPolynomialIdeal_magma_repr`

class `sage.rings.polynomial.multi_polynomial_ideal.``MPolynomialIdeal_singular_base_repr`
  `syzygy_module`()

    Computes the first syzygy (i.e., the module of relations of the given generators) of the ideal.

    EXAMPLE:

        sage: R.<x,y> = PolynomialRing(QQ)
        sage: f = 2*x^2 + y
        sage: g = y
        sage: h = 2*f + g
        sage: I = Ideal([f,g,h])
        sage: M = I.syzygy_module(); M
        [       -2        -1         1]
        [       -y 2*x^2 + y         0]
        sage: G = vector(I.gens())
        sage: M*G
        (0, 0)

    ALGORITHM: Uses Singular’s syz command

class `sage.rings.polynomial.multi_polynomial_ideal.``MPolynomialIdeal_singular_repr`

  Bases: `sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_base_repr`

  An ideal in a multivariate polynomial ring, which has an underlying Singular ring associated to it.

  `associated_primes`(algorithm='sy')

    Return a list of the associated primes of primary ideals of which the intersection is \(I\) = `self`.

    An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\) and if whenever \(ab \in Q\) and \(a \not\in Q\) then \(b^n \in Q\) for some \(n \in \ZZ\).

    If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\), i.e. \(P = \{a \in R, a^n \in Q\}\) for some \(n \in \ZZ\), is called the associated prime of \(Q\).

    If \(I\) is a proper ideal of the ring \(R\) then there exists a decomposition in primary ideals \(Q_i\) such that

    - their intersection is \(I\)
    - none of the \(Q_i\) contains the intersection of the rest, and
    - the associated prime ideals of \(Q_i\) are pairwise different.

    This method returns the associated primes of the \(Q_i\).

    INPUT:

    - `algorithm` - string:
    - `'sy'` - (default) use the shimoyama-yokoyama algorithm
    - `'gtz'` - use the gianni-trager-zacharias algorithm

    OUTPUT:

    - `list` - a list of associated primes

    EXAMPLES:

        sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
        sage: p = z^2 + 1; q = z^3 + 2
        sage: I = (p*q^2, y-z^2)*R
        sage: pd = I.associated_primes(); pd
        [Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
         Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field]

    ALGORITHM:

    Uses Singular.

    REFERENCES:

    - Thomas Becker and Volker Weispfenning. Groebner Bases - A Computational Approach To Commutative Algebra. Springer, New York 1993.
  `basis_is_groebner`(singular=Singular)

    Returns `True` if the generators of this ideal (`self.gens()`) form a Groebner basis.

    Let \(I\) be the set of generators of this ideal. The check is performed by trying to lift \(Syz(LM(I))\) to \(Syz(I)\) as \(I\) forms a Groebner basis if and only if for every element \(S\) in \(Syz(LM(I))\):

    \[\begin{split}S * G = \sum_{i=0}^{m} h_ig_i ---->_G 0.\end{split}\]

    ALGORITHM:

    Uses Singular.

    EXAMPLE:

        sage: R.<a,b,c,d,e,f,g,h,i,j> = PolynomialRing(GF(127),10)
        sage: I = sage.rings.ideal.Cyclic(R,4)
        sage: I.basis_is_groebner()
        False
        sage: I2 = Ideal(I.groebner_basis())
        sage: I2.basis_is_groebner()
        True

    A more complicated example:

        sage: R.<U6,U5,U4,U3,U2, u6,u5,u4,u3,u2, h> = PolynomialRing(GF(7583))
        sage: l = [u6 + u5 + u4 + u3 + u2 - 3791*h, \
                   U6 + U5 + U4 + U3 + U2 - 3791*h, \
                   U2*u2 - h^2, U3*u3 - h^2, U4*u4 - h^2, \
                   U5*u4 + U5*u3 + U4*u3 + U5*u2 + U4*u2 + U3*u2 - 3791*U5*h - 3791*U4*h - 3791*U3*h - 3791*U2*h - 2842*h^2, \
                   U4*u5 + U3*u5 + U2*u5 + U3*u4 + U2*u4 + U2*u3 - 3791*u5*h - 3791*u4*h - 3791*u3*h - 3791*u2*h - 2842*h^2, \
                   U5*u5 - h^2, U4*U2*u3 + U5*U3*u2 + U4*U3*u2 + U3^2*u2 - 3791*U5*U3*h - 3791*U4*U3*h - 3791*U3^2*h - 3791*U5*U2*h \
                    - 3791*U4*U2*h + U3*U2*h - 3791*U2^2*h - 3791*U4*u3*h - 3791*U4*u2*h - 3791*U3*u2*h - 2843*U5*h^2 + 1897*U4*h^2 - 946*U3*h^2 - 947*U2*h^2 + 2370*h^3, \
                   U3*u5*u4 + U2*u5*u4 + U3*u4^2 + U2*u4^2 + U2*u4*u3 - 3791*u5*u4*h - 3791*u4^2*h - 3791*u4*u3*h - 3791*u4*u2*h + u5*h^2 - 2842*u4*h^2, \
                   U2*u5*u4*u3 + U2*u4^2*u3 + U2*u4*u3^2 - 3791*u5*u4*u3*h - 3791*u4^2*u3*h - 3791*u4*u3^2*h - 3791*u4*u3*u2*h + u5*u4*h^2 + u4^2*h^2 + u5*u3*h^2 - 2842*u4*u3*h^2, \
                   U5^2*U4*u3 + U5*U4^2*u3 + U5^2*U4*u2 + U5*U4^2*u2 + U5^2*U3*u2 + 2*U5*U4*U3*u2 + U5*U3^2*u2 - 3791*U5^2*U4*h - 3791*U5*U4^2*h - 3791*U5^2*U3*h \
                    + U5*U4*U3*h - 3791*U5*U3^2*h - 3791*U5^2*U2*h + U5*U4*U2*h + U5*U3*U2*h - 3791*U5*U2^2*h - 3791*U5*U3*u2*h - 2842*U5^2*h^2 + 1897*U5*U4*h^2 \
                    - U4^2*h^2 - 947*U5*U3*h^2 - U4*U3*h^2 - 948*U5*U2*h^2 - U4*U2*h^2 - 1422*U5*h^3 + 3791*U4*h^3, \
                   u5*u4*u3*u2*h + u4^2*u3*u2*h + u4*u3^2*u2*h + u4*u3*u2^2*h + 2*u5*u4*u3*h^2 + 2*u4^2*u3*h^2 + 2*u4*u3^2*h^2 + 2*u5*u4*u2*h^2 + 2*u4^2*u2*h^2 \
                    + 2*u5*u3*u2*h^2 + 1899*u4*u3*u2*h^2, \
                   U5^2*U4*U3*u2 + U5*U4^2*U3*u2 + U5*U4*U3^2*u2 - 3791*U5^2*U4*U3*h - 3791*U5*U4^2*U3*h - 3791*U5*U4*U3^2*h - 3791*U5*U4*U3*U2*h \
                    + 3791*U5*U4*U3*u2*h + U5^2*U4*h^2 + U5*U4^2*h^2 + U5^2*U3*h^2 - U4^2*U3*h^2 - U5*U3^2*h^2 - U4*U3^2*h^2 - U5*U4*U2*h^2 \
                    - U5*U3*U2*h^2 - U4*U3*U2*h^2 + 3791*U5*U4*h^3 + 3791*U5*U3*h^3 + 3791*U4*U3*h^3, \
                   u4^2*u3*u2*h^2 + 1515*u5*u3^2*u2*h^2 + u4*u3^2*u2*h^2 + 1515*u5*u4*u2^2*h^2 + 1515*u5*u3*u2^2*h^2 + u4*u3*u2^2*h^2 \
                    + 1521*u5*u4*u3*h^3 - 3028*u4^2*u3*h^3 - 3028*u4*u3^2*h^3 + 1521*u5*u4*u2*h^3 - 3028*u4^2*u2*h^3 + 1521*u5*u3*u2*h^3 + 3420*u4*u3*u2*h^3, \
                   U5^2*U4*U3*U2*h + U5*U4^2*U3*U2*h + U5*U4*U3^2*U2*h + U5*U4*U3*U2^2*h + 2*U5^2*U4*U3*h^2 + 2*U5*U4^2*U3*h^2 + 2*U5*U4*U3^2*h^2 \
                    + 2*U5^2*U4*U2*h^2 + 2*U5*U4^2*U2*h^2 + 2*U5^2*U3*U2*h^2 - 2*U4^2*U3*U2*h^2 - 2*U5*U3^2*U2*h^2 - 2*U4*U3^2*U2*h^2 \
                     - 2*U5*U4*U2^2*h^2 - 2*U5*U3*U2^2*h^2 - 2*U4*U3*U2^2*h^2 - U5*U4*U3*h^3 - U5*U4*U2*h^3 - U5*U3*U2*h^3 - U4*U3*U2*h^3]

        sage: Ideal(l).basis_is_groebner()
        False
        sage: gb = Ideal(l).groebner_basis()
        sage: Ideal(gb).basis_is_groebner()
        True

    Note

    From the Singular Manual for the reduce function we use in this method: ‘The result may have no meaning if the second argument (`self`) is not a standard basis’. I (malb) believe this refers to the mathematical fact that the results may have no meaning if self is no standard basis, i.e., Singular doesn’t ‘add’ any additional ‘nonsense’ to the result. So we may actually use reduce to determine if self is a Groebner basis.

  `complete_primary_decomposition`(algorithm='sy')

    Return a list of primary ideals such that their intersection is `self`, together with the associated prime ideals.

    An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\), and if whenever \(ab \in Q\) and \(a \not\in Q\), then \(b^n \in Q\) for some \(n \in \ZZ\).

    If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\) (i.e. the ideal consisting of all \(a \in R\) with a^n in Q` for some \(n \in \ZZ\)), is called the associated prime of \(Q\).

    If \(I\) is a proper ideal of a Noetherian ring \(R\), then there exists a finite collection of primary ideals \(Q_i\) such that the following hold:

    - the intersection of the \(Q_i\) is \(I\);
    - none of the \(Q_i\) contains the intersection of the others;
    - the associated prime ideals \(P_i\) of the \(Q_i\) are pairwise distinct.

    INPUT:

    - `algorithm` – string:
      - `'sy'` – (default) use the Shimoyama-Yokoyama algorithm
      - `'gtz'` – use the Gianni-Trager-Zacharias algorithm

    OUTPUT:

    - a list of pairs \((Q_i, P_i)\), where the \(Q_i\) form a primary decomposition of `self` and \(P_i\) is the associated prime of \(Q_i\).

    EXAMPLES:

        sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
        sage: p = z^2 + 1; q = z^3 + 2
        sage: I = (p*q^2, y-z^2)*R
        sage: pd = I.complete_primary_decomposition(); pd
        [(Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field),
         (Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field)]

        sage: I.primary_decomposition_complete(algorithm = 'gtz')
        [(Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field),
         (Ideal (z^2 + 1, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^2 + 1, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field)]

        sage: from functools import reduce
        sage: reduce(lambda Qi,Qj: Qi.intersection(Qj), [Qi for (Qi,radQi) in pd]) == I
        True

        sage: [Qi.radical() == radQi for (Qi,radQi) in pd]
        [True, True]

        sage: P.<x,y,z> = PolynomialRing(ZZ)
        sage: I = ideal( x^2 - 3*y, y^3 - x*y, z^3 - x, x^4 - y*z + 1 )
        sage: I.complete_primary_decomposition()
        Traceback (most recent call last):
        ...
        ValueError: Coefficient ring must be a field for function 'complete_primary_decomposition'.

    ALGORITHM:

    Uses Singular.

    Note

    See [BW93] for an introduction to primary decomposition.

    TESTS:

    Check that trac ticket #15745 is fixed:

        sage: R.<x,y>= QQ[]
        sage: I = Ideal(R(1))
        sage: I.complete_primary_decomposition()
        []
        sage: I.is_prime()
        False

  `dimension`(singular='singular_default')

    The dimension of the ring modulo this ideal.

    EXAMPLE:

        sage: P.<x,y,z> = PolynomialRing(GF(32003),order='degrevlex')
        sage: I = ideal(x^2-y,x^3)
        sage: I.dimension()
        1

    For polynomials over a finite field of order too large for Singular, this falls back on a toy implementation of Buchberger to compute the Groebner basis, then uses the algorithm described in Chapter 9, Section 1 of Cox, Little, and O’Shea’s “Ideals, Varieties, and Algorithms”.

    EXAMPLE:

        sage: R.<x,y> = PolynomialRing(GF(2147483659),order='lex')
        sage: I = R.ideal([x*y,x*y+1])
        sage: I.dimension()
        verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
        0
        sage: I=ideal([x*(x*y+1),y*(x*y+1)])
        sage: I.dimension()
        verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
        1
        sage: I = R.ideal([x^3*y,x*y^2])
        sage: I.dimension()
        verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
        1
        sage: R.<x,y> = PolynomialRing(GF(2147483659),order='lex')
        sage: I = R.ideal(0)
        sage: I.dimension()
        verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
        2

    ALGORITHM:

    Uses Singular, unless the characteristic is too large.

    Note

    Requires computation of a Groebner basis, which can be a very expensive operation.

  `elimination_ideal`(variables)

    Returns the elimination ideal this ideal with respect to the variables given in `variables`.

    INPUT:

    - `variables` - a list or tuple of variables in `self.ring()`

    EXAMPLE:

        sage: R.<x,y,t,s,z> = PolynomialRing(QQ,5)
        sage: I = R * [x-t,y-t^2,z-t^3,s-x+y^3]
        sage: I.elimination_ideal([t,s])
        Ideal (y^2 - x*z, x*y - z, x^2 - y) of Multivariate
        Polynomial Ring in x, y, t, s, z over Rational Field

    ALGORITHM:

    Uses Singular.

    Note

    Requires computation of a Groebner basis, which can be a very expensive operation.

  `genus`()

    Return the genus of the projective curve defined by this ideal, which must be 1 dimensional.

    EXAMPLE:

    Consider the hyperelliptic curve \(y^2 = 4x^5 - 30x^3 + 45x - 22\) over \(\QQ\), it has genus 2:

        sage: P, x = PolynomialRing(QQ,"x").objgen()
        sage: f = 4*x^5 - 30*x^3 + 45*x - 22
        sage: C = HyperellipticCurve(f); C
        Hyperelliptic Curve over Rational Field defined by y^2 = 4*x^5 - 30*x^3 + 45*x - 22
        sage: C.genus()
        2

        sage: P.<x,y> = PolynomialRing(QQ)
        sage: f = y^2 - 4*x^5 - 30*x^3 + 45*x - 22
        sage: I = Ideal([f])
        sage: I.genus()
        2

    TESTS:

    Check that the answer is correct for reducible curves:

        sage: R.<x, y, z> = QQ[]
        sage: C = Curve(x^2 - 2*y^2)
        sage: C.is_singular()
        True
        sage: C.genus()
        -1
        sage: Ideal(x^4+y^2*x+x).genus()
        0
        sage: T.<t1,t2,u1,u2> = QQ[]
        sage: TJ = Ideal([t1^2 + u1^2 - 1,t2^2 + u2^2 - 1, (t1-t2)^2 + (u1-u2)^2 -1])
        sage: TJ.genus()
        -1

  `hilbert_polynomial`()

    Return the Hilbert polynomial of this ideal.

    Let \(I\) = `self` be a homogeneous ideal and \(R\) = `self.ring()` be a graded commutative algebra (\(R = \oplus R_d\)) over a field \(K\). The Hilbert polynomial is the unique polynomial \(HP(t)\) with rational coefficients such that \(HP(d) = dim_K R_d\) for all but finitely many positive integers \(d\).

    EXAMPLE:

        sage: P.<x,y,z> = PolynomialRing(QQ)
        sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
        sage: I.hilbert_polynomial()
        5*t - 5

  `hilbert_series`(singular='singular_default')

    Return the Hilbert series of this ideal.

    Let \(I\) = `self` be a homogeneous ideal and \(R\) = `self.ring()` be a graded commutative algebra (\(R = \oplus R_d\)) over a field \(K\). Then the Hilbert function is defined as \(H(d) = dim_K R_d\) and the Hilbert series of \(I\) is defined as the formal power series \(HS(t) = \sum_0^{\infty} H(d) t^d\).

    This power series can be expressed as \(HS(t) = Q(t)/(1-t)^n\) where \(Q(t)\) is a polynomial over \(Z\) and \(n\) the number of variables in \(R\). This method returns \(Q(t)/(1-t)^n\).

    EXAMPLE:

        sage: P.<x,y,z> = PolynomialRing(QQ)
        sage: I = Ideal([x^3*y^2 + 3*x^2*y^2*z + y^3*z^2 + z^5])
        sage: I.hilbert_series()
        (-t^4 - t^3 - t^2 - t - 1)/(-t^2 + 2*t - 1)

  `integral_closure`(p=0, r=True, singular='singular_default')

    Let \(I\) = `self`.

    Returns the integral closure of \(I, ..., I^p\), where \(sI\) is an ideal in the polynomial ring \(R=k[x(1),...x(n)]\). If \(p\) is not given, or \(p=0\), compute the closure of all powers up to the maximum degree in t occurring in the closure of \(R[It]\) (so this is the last power whose closure is not just the sum/product of the smaller). If \(r\) is given and `r is True`, `I.integral_closure()` starts with a check whether I is already a radical ideal.

    INPUT:

    - `p` - powers of I (default: 0)
    - `r` - check whether self is a radical ideal first (default: `True`)

    EXAMPLE:

        sage: R.<x,y> = QQ[]
        sage: I = ideal([x^2,x*y^4,y^5])
        sage: I.integral_closure()
        [x^2, y^5, -x*y^3]

    ALGORITHM:

    Uses libSINGULAR.

  `interreduced_basis`()

    If this ideal is spanned by \((f_1, ..., f_n)\) this method returns \((g_1, ..., g_s)\) such that:

    - \((f_1,...,f_n) = (g_1,...,g_s)\)

    - \(LT(g_i) != LT(g_j)\) for all \(i != j\)

    - \(LT(g_i)\) does not divide \(m\) for all monomials \(m\) of

        \(\{g_1,...,g_{i-1},g_{i+1},...,g_s\}\)

    - \(LC(g_i) == 1\) for all \(i\) if the coefficient ring is a field.

    EXAMPLE:

        sage: R.<x,y,z> = PolynomialRing(QQ)
        sage: I = Ideal([z*x+y^3,z+y^3,z+x*y])
        sage: I.interreduced_basis()
        [y^3 + z, x*y + z, x*z - z]

    Note that tail reduction for local orderings is not well-defined:

        sage: R.<x,y,z> = PolynomialRing(QQ,order='negdegrevlex')
        sage: I = Ideal([z*x+y^3,z+y^3,z+x*y])
        sage: I.interreduced_basis()
        [z + x*y, x*y - y^3, x^2*y - y^3]

    A fixed error with nonstandard base fields:

        sage: R.<t>=QQ['t']
        sage: K.<x,y>=R.fraction_field()['x,y']
        sage: I=t*x*K
        sage: I.interreduced_basis()
        [x]

    The interreduced basis of 0 is 0:

        sage: P.<x,y,z> = GF(2)[]
        sage: Ideal(P(0)).interreduced_basis()
        [0]

    ALGORITHM:

    Uses Singular’s interred command or `sage.rings.polynomial.toy_buchberger.inter_reduction()` if conversion to Singular fails.

  `intersection`(*others)

    Return the intersection of the arguments with this ideal.

    EXAMPLES:

        sage: R.<x,y> = PolynomialRing(QQ, 2, order='lex')
        sage: I = x*R
        sage: J = y*R
        sage: I.intersection(J)
        Ideal (x*y) of Multivariate Polynomial Ring in x, y over Rational Field

    The following simple example illustrates that the product need not equal the intersection.

        sage: I = (x^2, y)*R
        sage: J = (y^2, x)*R
        sage: K = I.intersection(J); K
        Ideal (y^2, x*y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field
        sage: IJ = I*J; IJ
        Ideal (x^2*y^2, x^3, y^3, x*y) of Multivariate Polynomial Ring in x, y over Rational Field
        sage: IJ == K
        False

    Intersection of several ideals:

        sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
        sage: I1 = x*R
        sage: I2 = y*R
        sage: I3 = (x, y)*R
        sage: I4 = (x^2 + x*y*z, y^2 - z^3*y, z^3 + y^5*x*z)*R
        sage: I1.intersection(I2, I3, I4)
        Ideal (x*y*z^20 - x*y*z^3, x*y^2 - x*y*z^3, x^2*y + x*y*z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field

    The ideals must share the same ring:

        sage: R2.<x,y> = PolynomialRing(QQ, 2, order='lex')
        sage: R3.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
        sage: I2 = x*R2
        sage: I3 = x*R3
        sage: I2.intersection(I3)
        Traceback (most recent call last):
        ...
        TypeError: Intersection is only available for ideals of the same ring.

  `is_prime`(**kwds)

    Return `True` if this ideal is prime.

    INPUT:

    - keyword arguments are passed on to `complete_primary_decomposition`; in this way you can specify the algorithm to use.

    EXAMPLES:

        sage: R.<x, y> = PolynomialRing(QQ, 2)
        sage: I = (x^2 - y^2 - 1)*R
        sage: I.is_prime()
        True
        sage: (I^2).is_prime()
        False

        sage: J = (x^2 - y^2)*R
        sage: J.is_prime()
        False
        sage: (J^3).is_prime()
        False

        sage: (I * J).is_prime()
        False

    The following is trac ticket #5982. Note that the quotient ring is not recognized as being a field at this time, so the fraction field is not the quotient ring itself:

        sage: Q = R.quotient(I); Q
        Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 - y^2 - 1)
        sage: Q.fraction_field()
        Fraction Field of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 - y^2 - 1)

  `minimal_associated_primes`()

    OUTPUT:

    - `list` - a list of prime ideals

    EXAMPLES:

        sage: R.<x,y,z> = PolynomialRing(QQ, 3, 'xyz')
        sage: p = z^2 + 1; q = z^3 + 2
        sage: I = (p*q^2, y-z^2)*R
        sage: I.minimal_associated_primes ()
        [Ideal (z^2 + 1, -z^2 + y) of Multivariate Polynomial Ring
        in x, y, z over Rational Field, Ideal (z^3 + 2, -z^2 + y)
        of Multivariate Polynomial Ring in x, y, z over Rational
        Field]

    ALGORITHM:

    Uses Singular.

  `normal_basis`(algorithm='libsingular', singular='singular_default')

    Returns a vector space basis (consisting of monomials) of the quotient ring by the ideal, resp. of a free module by the module, in case it is finite dimensional and if the input is a standard basis with respect to the ring ordering.

    INPUT:

    `algorithm` - defaults to use libsingular, if it is anything else we will use the `kbase()` command

    EXAMPLES:

        sage: R.<x,y,z> = PolynomialRing(QQ)
        sage: I = R.ideal(x^2+y^2+z^2-4, x^2+2*y^2-5, x*z-1)
        sage: I.normal_basis()
        [y*z^2, z^2, y*z, z, x*y, y, x, 1]
        sage: I.normal_basis(algorithm='singular')
        [y*z^2, z^2, y*z, z, x*y, y, x, 1]

  `plot`(singular=Singular)

    If you somehow manage to install surf, perhaps you can use this function to implicitly plot the real zero locus of this ideal (if principal).

    INPUT:

    - `self` - must be a principal ideal in 2 or 3 vars over \(\QQ\).

    EXAMPLES:

    Implicit plotting in 2-d:

        sage: R.<x,y> = PolynomialRing(QQ,2)
        sage: I = R.ideal([y^3 - x^2])
        sage: I.plot()        # cusp
        Graphics object consisting of 1 graphics primitive
        sage: I = R.ideal([y^2 - x^2 - 1])
        sage: I.plot()        # hyperbola
        Graphics object consisting of 1 graphics primitive
        sage: I = R.ideal([y^2 + x^2*(1/4) - 1])
        sage: I.plot()        # ellipse
        Graphics object consisting of 1 graphics primitive
        sage: I = R.ideal([y^2-(x^2-1)*(x-2)])
        sage: I.plot()        # elliptic curve
        Graphics object consisting of 1 graphics primitive

    Implicit plotting in 3-d:

        sage: R.<x,y,z> = PolynomialRing(QQ,3)
        sage: I = R.ideal([y^2 + x^2*(1/4) - z])
        sage: I.plot()          # a cone; optional - surf
        sage: I = R.ideal([y^2 + z^2*(1/4) - x])
        sage: I.plot()          # same code, from a different angle; optional - surf
        sage: I = R.ideal([x^2*y^2+x^2*z^2+y^2*z^2-16*x*y*z])
        sage: I.plot()          # Steiner surface; optional - surf

    AUTHORS:

    - David Joyner (2006-02-12)
  `primary_decomposition`(algorithm='sy')

    Return a list of primary ideals such that their intersection is `self`.

    An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\), and if whenever \(ab \in Q\) and \(a \not\in Q\), then \(b^n \in Q\) for some \(n \in \ZZ\).

    If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\) (i.e. the ideal consisting of all \(a \in R\) with a^n in Q` for some \(n \in \ZZ\)), is called the associated prime of \(Q\).

    If \(I\) is a proper ideal of a Noetherian ring \(R\), then there exists a finite collection of primary ideals \(Q_i\) such that the following hold:

    - the intersection of the \(Q_i\) is \(I\);
    - none of the \(Q_i\) contains the intersection of the others;
    - the associated prime ideals of the \(Q_i\) are pairwise distinct.

    INPUT:

    - `algorithm` – string:
      - `'sy'` – (default) use the Shimoyama-Yokoyama algorithm
      - `'gtz'` – use the Gianni-Trager-Zacharias algorithm

    OUTPUT:

    - a list of primary ideals \(Q_i\) forming a primary decomposition of `self`.

    EXAMPLES:

        sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
        sage: p = z^2 + 1; q = z^3 + 2
        sage: I = (p*q^2, y-z^2)*R
        sage: pd = I.primary_decomposition(); pd
        [Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
         Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field]

        sage: from functools import reduce
        sage: reduce(lambda Qi,Qj: Qi.intersection(Qj), pd) == I
        True

    ALGORITHM:

    Uses Singular.

    REFERENCES:

    - Thomas Becker and Volker Weispfenning. Groebner Bases - A Computational Approach To Commutative Algebra. Springer, New York 1993.
  `primary_decomposition_complete`(algorithm='sy')

    Return a list of primary ideals such that their intersection is `self`, together with the associated prime ideals.

    An ideal \(Q\) is called primary if it is a proper ideal of the ring \(R\), and if whenever \(ab \in Q\) and \(a \not\in Q\), then \(b^n \in Q\) for some \(n \in \ZZ\).

    If \(Q\) is a primary ideal of the ring \(R\), then the radical ideal \(P\) of \(Q\) (i.e. the ideal consisting of all \(a \in R\) with a^n in Q` for some \(n \in \ZZ\)), is called the associated prime of \(Q\).

    If \(I\) is a proper ideal of a Noetherian ring \(R\), then there exists a finite collection of primary ideals \(Q_i\) such that the following hold:

    - the intersection of the \(Q_i\) is \(I\);
    - none of the \(Q_i\) contains the intersection of the others;
    - the associated prime ideals \(P_i\) of the \(Q_i\) are pairwise distinct.

    INPUT:

    - `algorithm` – string:
      - `'sy'` – (default) use the Shimoyama-Yokoyama algorithm
      - `'gtz'` – use the Gianni-Trager-Zacharias algorithm

    OUTPUT:

    - a list of pairs \((Q_i, P_i)\), where the \(Q_i\) form a primary decomposition of `self` and \(P_i\) is the associated prime of \(Q_i\).

    EXAMPLES:

        sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
        sage: p = z^2 + 1; q = z^3 + 2
        sage: I = (p*q^2, y-z^2)*R
        sage: pd = I.complete_primary_decomposition(); pd
        [(Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field),
         (Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field)]

        sage: I.primary_decomposition_complete(algorithm = 'gtz')
        [(Ideal (z^6 + 4*z^3 + 4, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field),
         (Ideal (z^2 + 1, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
          Ideal (z^2 + 1, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field)]

        sage: from functools import reduce
        sage: reduce(lambda Qi,Qj: Qi.intersection(Qj), [Qi for (Qi,radQi) in pd]) == I
        True

        sage: [Qi.radical() == radQi for (Qi,radQi) in pd]
        [True, True]

        sage: P.<x,y,z> = PolynomialRing(ZZ)
        sage: I = ideal( x^2 - 3*y, y^3 - x*y, z^3 - x, x^4 - y*z + 1 )
        sage: I.complete_primary_decomposition()
        Traceback (most recent call last):
        ...
        ValueError: Coefficient ring must be a field for function 'complete_primary_decomposition'.

    ALGORITHM:

    Uses Singular.

    Note

    See [BW93] for an introduction to primary decomposition.

    TESTS:

    Check that trac ticket #15745 is fixed:

        sage: R.<x,y>= QQ[]
        sage: I = Ideal(R(1))
        sage: I.complete_primary_decomposition()
        []
        sage: I.is_prime()
        False

  `quotient`(J)

    Given ideals \(I\) = `self` and \(J\) in the same polynomial ring \(P\), return the ideal quotient of \(I\) by \(J\) consisting of the polynomials \(a\) of \(P\) such that \(\{aJ \subset I\}\).

    This is also referred to as the colon ideal (\(I\):\(J\)).

    INPUT:

    - `J` - multivariate polynomial ideal

    EXAMPLE:

        sage: R.<x,y,z> = PolynomialRing(GF(181),3)
        sage: I = Ideal([x^2+x*y*z,y^2-z^3*y,z^3+y^5*x*z])
        sage: J = Ideal([x])
        sage: Q = I.quotient(J)
        sage: y*z + x in I
        False
        sage: x in J
        True
        sage: x * (y*z + x) in I
        True

  `radical`()

    The radical of this ideal.

    EXAMPLES:

    This is an obviously not radical ideal:

        sage: R.<x,y,z> = PolynomialRing(QQ, 3)
        sage: I = (x^2, y^3, (x*z)^4 + y^3 + 10*x^2)*R
        sage: I.radical()
        Ideal (y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field

    That the radical is correct is clear from the Groebner basis.

        sage: I.groebner_basis()
        [y^3, x^2]

    This is the example from the Singular manual:

        sage: p = z^2 + 1; q = z^3 + 2
        sage: I = (p*q^2, y-z^2)*R
        sage: I.radical()
        Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Multivariate Polynomial Ring in x, y, z over Rational Field

    Note

    From the Singular manual: A combination of the algorithms of Krick/Logar and Kemper is used. Works also in positive characteristic (Kempers algorithm).

        sage: R.<x,y,z> = PolynomialRing(GF(37), 3)
        sage: p = z^2 + 1; q = z^3 + 2
        sage: I = (p*q^2, y - z^2)*R
        sage: I.radical()
        Ideal (z^2 - y, y^2*z + y*z + 2*y + 2) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 37

  `saturation`(other)

    Returns the saturation (and saturation exponent) of the ideal `self` with respect to the ideal `other`

    INPUT:

    - `other` – another ideal in the same ring

    OUTPUT:

    - a pair (ideal, integer)

    EXAMPLES:

        sage: R.<x, y, z> = QQ[]
        sage: I = R.ideal(x^5*z^3, x*y*z, y*z^4)
        sage: J = R.ideal(z)
        sage: I.saturation(J)
        (Ideal (y, x^5) of Multivariate Polynomial Ring in x, y, z over Rational Field, 4)

  `syzygy_module`()

    Computes the first syzygy (i.e., the module of relations of the given generators) of the ideal.

    EXAMPLE:

        sage: R.<x,y> = PolynomialRing(QQ)
        sage: f = 2*x^2 + y
        sage: g = y
        sage: h = 2*f + g
        sage: I = Ideal([f,g,h])
        sage: M = I.syzygy_module(); M
        [       -2        -1         1]
        [       -y 2*x^2 + y         0]
        sage: G = vector(I.gens())
        sage: M*G
        (0, 0)

    ALGORITHM:

    Uses Singular’s syz command.

  `transformed_basis`(algorithm='gwalk', other_ring=None, singular='singular_default')

    Returns a lex or `other_ring` Groebner Basis for this ideal.

    INPUT:

    - `algorithm` - see below for options.

    - `other_ring` - only valid for algorithm ‘fglm’, if

        provided conversion will be performed to this ring. Otherwise a lex Groebner basis will be returned.

    ALGORITHMS:

    fglm
      FGLM algorithm. The input ideal must be given with a reduced Groebner Basis of a zero-dimensional ideal
    gwalk
      Groebner Walk algorithm (default)
    awalk1
      ‘first alternative’ algorithm
    awalk2
      ‘second alternative’ algorithm
    twalk
      Tran algorithm
    fwalk
      Fractal Walk algorithm

    EXAMPLES:

        sage: R.<x,y,z> = PolynomialRing(QQ,3)
        sage: I = Ideal([y^3+x^2,x^2*y+x^2, x^3-x^2, z^4-x^2-y])
        sage: I = Ideal(I.groebner_basis())
        sage: S.<z,x,y> = PolynomialRing(QQ,3,order='lex')
        sage: J = Ideal(I.transformed_basis('fglm',S))
        sage: J
        Ideal (z^4 + y^3 - y, x^2 + y^3, x*y^3 - y^3, y^4 + y^3)
         of Multivariate Polynomial Ring in z, x, y over Rational Field

        sage: R.<z,y,x>=PolynomialRing(GF(32003),3,order='lex')
        sage: I=Ideal([y^3+x*y*z+y^2*z+x*z^3,3+x*y+x^2*y+y^2*z])
        sage: I.transformed_basis('gwalk')
        [z*y^2 + y*x^2 + y*x + 3,
         z*x + 8297*y^8*x^2 + 8297*y^8*x + 3556*y^7 - 8297*y^6*x^4 + 15409*y^6*x^3 - 8297*y^6*x^2
         - 8297*y^5*x^5 + 15409*y^5*x^4 - 8297*y^5*x^3 + 3556*y^5*x^2 + 3556*y^5*x + 3556*y^4*x^3
         + 3556*y^4*x^2 - 10668*y^4 - 10668*y^3*x - 8297*y^2*x^9 - 1185*y^2*x^8 + 14224*y^2*x^7
         - 1185*y^2*x^6 - 8297*y^2*x^5 - 14223*y*x^7 - 10666*y*x^6 - 10666*y*x^5 - 14223*y*x^4
         + x^5 + 2*x^4 + x^3,
         y^9 - y^7*x^2 - y^7*x - y^6*x^3 - y^6*x^2 - 3*y^6 - 3*y^5*x - y^3*x^7 - 3*y^3*x^6
         - 3*y^3*x^5 - y^3*x^4 - 9*y^2*x^5 - 18*y^2*x^4 - 9*y^2*x^3 - 27*y*x^3 - 27*y*x^2 - 27*x]

    ALGORITHM:

    Uses Singular.

  `triangular_decomposition`(algorithm=None, singular='singular_default')

    Decompose zero-dimensional ideal `self` into triangular sets.

    This requires that the given basis is reduced w.r.t. to the lexicographical monomial ordering. If the basis of self does not have this property, the required Groebner basis is computed implicitly.

    INPUT:

    - `algorithm` - string or None (default: None)

    ALGORITHMS:

    - `singular:triangL` - decomposition of self into triangular systems (Lazard).
    - `singular:triangLfak` - decomp. of self into tri. systems plus factorization.
      - `singular:triangM` - decomposition of self into triangular systems (Moeller).

    OUTPUT: a list \(T\) of lists \(t\) such that the variety of `self` is the union of the varieties of \(t\) in \(L\) and each \(t\) is in triangular form.

    EXAMPLE:

        sage: P.<e,d,c,b,a> = PolynomialRing(QQ,5,order='lex')
        sage: I = sage.rings.ideal.Cyclic(P)
        sage: GB = Ideal(I.groebner_basis('libsingular:stdfglm'))
        sage: GB.triangular_decomposition('singular:triangLfak')
        [Ideal (a - 1, b - 1, c - 1, d^2 + 3*d + 1, e + d + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a - 1, b - 1, c^2 + 3*c + 1, d + c + 3, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a - 1, b^2 + 3*b + 1, c + b + 3, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a - 1, b^4 + b^3 + b^2 + b + 1, -c + b^2, -d + b^3, e + b^3 + b^2 + b + 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^2 + 3*a + 1, b - 1, c - 1, d - 1, e + a + 3) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^2 + 3*a + 1, b + a + 3, c - 1, d - 1, e - 1) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^4 - 4*a^3 + 6*a^2 + a + 1, -11*b^2 + 6*b*a^3 - 26*b*a^2 + 41*b*a - 4*b - 8*a^3 + 31*a^2 - 40*a - 24, 11*c + 3*a^3 - 13*a^2 + 26*a - 2, 11*d + 3*a^3 - 13*a^2 + 26*a - 2, -11*e - 11*b + 6*a^3 - 26*a^2 + 41*a - 4) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^4 + a^3 + a^2 + a + 1, b - 1, c + a^3 + a^2 + a + 1, -d + a^3, -e + a^2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^4 + a^3 + a^2 + a + 1, b - a, c - a, d^2 + 3*d*a + a^2, e + d + 3*a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^4 + a^3 + a^2 + a + 1, b - a, c^2 + 3*c*a + a^2, d + c + 3*a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^4 + a^3 + a^2 + a + 1, b^2 + 3*b*a + a^2, c + b + 3*a, d - a, e - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^4 + a^3 + a^2 + a + 1, b^3 + b^2*a + b^2 + b*a^2 + b*a + b + a^3 + a^2 + a + 1, c + b^2*a^3 + b^2*a^2 + b^2*a + b^2, -d + b^2*a^2 + b^2*a + b^2 + b*a^2 + b*a + a^2, -e + b^2*a^3 - b*a^2 - b*a - b - a^2 - a) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field,
        Ideal (a^4 + a^3 + 6*a^2 - 4*a + 1, -11*b^2 + 6*b*a^3 + 10*b*a^2 + 39*b*a + 2*b + 16*a^3 + 23*a^2 + 104*a - 24, 11*c + 3*a^3 + 5*a^2 + 25*a + 1, 11*d + 3*a^3 + 5*a^2 + 25*a + 1, -11*e - 11*b + 6*a^3 + 10*a^2 + 39*a + 2) of Multivariate Polynomial Ring in e, d, c, b, a over Rational Field]

        sage: R.<x1,x2> = PolynomialRing(QQ, 2, order='lex')
        sage: f1 = 1/2*((x1^2 + 2*x1 - 4)*x2^2 + 2*(x1^2 + x1)*x2 + x1^2)
        sage: f2 = 1/2*((x1^2 + 2*x1 + 1)*x2^2 + 2*(x1^2 + x1)*x2 - 4*x1^2)
        sage: I = Ideal(f1,f2)
        sage: I.triangular_decomposition()
        [Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
         Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
         Ideal (x2, x1^2) of Multivariate Polynomial Ring in x1, x2 over Rational Field,
         Ideal (x2^4 + 4*x2^3 - 6*x2^2 - 20*x2 + 5, 8*x1 - x2^3 + x2^2 + 13*x2 - 5) of Multivariate Polynomial Ring in x1, x2 over Rational Field]

    TESTS:

        sage: R.<x,y> = QQ[]
        sage: J = Ideal(x^2+y^2-2, y^2-1)
        sage: J.triangular_decomposition()
        [Ideal (y^2 - 1, x^2 - 1) of Multivariate Polynomial Ring in x, y over Rational Field]

  `variety`(ring=None)

    Return the variety of this ideal.

    Given a zero-dimensional ideal \(I\) (== `self`) of a polynomial ring \(P\) whose order is lexicographic, return the variety of \(I\) as a list of dictionaries with `(variable, value)` pairs. By default, the variety of the ideal over its coefficient field \(K\) is returned; `ring` can be specified to find the variety over a different ring.

    These dictionaries have cardinality equal to the number of variables in \(P\) and represent assignments of values to these variables such that all polynomials in \(I\) vanish.

    If `ring` is specified, then a triangular decomposition of `self` is found over the original coefficient field \(K\); then the triangular systems are solved using root-finding over `ring`. This is particularly useful when \(K\) is `QQ` (to allow fast symbolic computation of the triangular decomposition) and `ring` is `RR`, `AA`, `CC`, or `QQbar` (to compute the whole real or complex variety of the ideal).

    Note that with `ring=RR` or `CC`, computation is done numerically and potentially inaccurately; in particular, the number of points in the real variety may be miscomputed. With `ring=AA` or `QQbar`, computation is done exactly (which may be much slower, of course).

    INPUT:

    - `ring` - return roots in the `ring` instead of the base ring of this ideal (default: `None`)
    - `proof` - return a provably correct result (default: `True`)

    EXAMPLE:

        sage: K.<w> = GF(27) # this example is from the MAGMA handbook
        sage: P.<x, y> = PolynomialRing(K, 2, order='lex')
        sage: I = Ideal([ x^8 + y + 2, y^6 + x*y^5 + x^2 ])
        sage: I = Ideal(I.groebner_basis()); I
        Ideal (x - y^47 - y^45 + y^44 - y^43 + y^41 - y^39 - y^38
        - y^37 - y^36 + y^35 - y^34 - y^33 + y^32 - y^31 + y^30 +
        y^28 + y^27 + y^26 + y^25 - y^23 + y^22 + y^21 - y^19 -
        y^18 - y^16 + y^15 + y^13 + y^12 - y^10 + y^9 + y^8 + y^7
        - y^6 + y^4 + y^3 + y^2 + y - 1, y^48 + y^41 - y^40 + y^37
        - y^36 - y^33 + y^32 - y^29 + y^28 - y^25 + y^24 + y^2 + y
        + 1) of Multivariate Polynomial Ring in x, y over Finite
        Field in w of size 3^3

        sage: V = I.variety(); V
        [{y: w^2 + 2, x: 2*w}, {y: w^2 + w, x: 2*w + 1}, {y: w^2 + 2*w, x: 2*w + 2}]

        sage: [f.subs(v) for f in I.gens() for v in V] # check that all polynomials vanish
        [0, 0, 0, 0, 0, 0]
        sage: [I.subs(v).is_zero() for v in V] # same test, but nicer syntax
        [True, True, True]

    However, we only account for solutions in the ground field and not in the algebraic closure:

        sage: I.vector_space_dimension()
        48

    Here we compute the points of intersection of a hyperbola and a circle, in several fields:

        sage: K.<x, y> = PolynomialRing(QQ, 2, order='lex')
        sage: I = Ideal([ x*y - 1, (x-2)^2 + (y-1)^2 - 1])
        sage: I = Ideal(I.groebner_basis()); I
        Ideal (x + y^3 - 2*y^2 + 4*y - 4, y^4 - 2*y^3 + 4*y^2 - 4*y + 1)
        of Multivariate Polynomial Ring in x, y over Rational Field

    These two curves have one rational intersection:

        sage: I.variety()
        [{y: 1, x: 1}]

    There are two real intersections:

        sage: I.variety(ring=RR)
        [{y: 0.361103080528647, x: 2.76929235423863},
         {y: 1.00000000000000, x: 1.00000000000000}]
        sage: I.variety(ring=AA)
        [{x: 2.769292354238632?, y: 0.3611030805286474?},
         {x: 1, y: 1}]

    and a total of four intersections:

        sage: I.variety(ring=CC)
        [{y: 0.31944845973567... - 1.6331702409152...*I,
          x: 0.11535382288068... + 0.58974280502220...*I},
         {y: 0.31944845973567... + 1.6331702409152...*I,
          x: 0.11535382288068... - 0.58974280502220...*I},
         {y: 0.36110308052864..., x: 2.7692923542386...},
         {y: 1.00000000000000, x: 1.00000000000000}]
        sage: I.variety(ring=QQbar)
        [{y: 0.3194484597356763? - 1.633170240915238?*I,
          x: 0.11535382288068429? + 0.5897428050222055?*I},
         {y: 0.3194484597356763? + 1.633170240915238?*I,
          x: 0.11535382288068429? - 0.5897428050222055?*I},
         {y: 0.3611030805286474?, x: 2.769292354238632?},
         {y: 1, x: 1}]

    Computation over floating point numbers may compute only a partial solution, or even none at all. Notice that x values are missing from the following variety:

        sage: R.<x,y> = CC[]
        sage: I = ideal([x^2+y^2-1,x*y-1])
        sage: I.variety()
        verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: computations in the complex field are inexact; variety may be computed partially or incorrectly.
        verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
        [{y: -0.86602540378443... - 0.500000000000000*I},
         {y: -0.86602540378443... + 0.500000000000000*I},
         {y: 0.86602540378443... - 0.500000000000000*I},
         {y: 0.86602540378443... + 0.500000000000000*I}]

    This is due to precision error, which causes the computation of an intermediate Groebner basis to fail.

    If the ground field’s characteristic is too large for Singular, we resort to a toy implementation:

        sage: R.<x,y> = PolynomialRing(GF(2147483659),order='lex')
        sage: I=ideal([x^3-2*y^2,3*x+y^4])
        sage: I.variety()
        verbose 0 (...: multi_polynomial_ideal.py, groebner_basis) Warning: falling back to very slow toy implementation.
        verbose 0 (...: multi_polynomial_ideal.py, dimension) Warning: falling back to very slow toy implementation.
        verbose 0 (...: multi_polynomial_ideal.py, variety) Warning: falling back to very slow toy implementation.
        [{y: 0, x: 0}]

    TESTS:

        sage: K.<w> = GF(27)
        sage: P.<x, y> = PolynomialRing(K, 2, order='lex')
        sage: I = Ideal([ x^8 + y + 2, y^6 + x*y^5 + x^2 ])

    Testing the robustness of the Singular interface:

        sage: T = I.triangular_decomposition('singular:triangLfak')
        sage: I.variety()
        [{y: w^2 + 2, x: 2*w}, {y: w^2 + w, x: 2*w + 1}, {y: w^2 + 2*w, x: 2*w + 2}]

    Testing that a bug is indeed fixed

        sage: R = PolynomialRing(GF(2), 30, ['x%d'%(i+1) for i in range(30)], order='lex')
        sage: R.inject_variables()
        Defining...
        sage: I = Ideal([x1 + 1, x2, x3 + 1, x5*x10 + x10 + x18, x5*x11 + x11, \
                         x5*x18, x6, x7 + 1, x9, x10*x11 + x10 + x18, x10*x18 + x18, \
                         x11*x18, x12, x13, x14, x15, x16 + 1, x17 + x18 + 1, x19, x20, \
                         x21 + 1, x22, x23, x24, x25 + 1, x28 + 1, x29 + 1, x30, x8, \
                         x26, x1^2 + x1, x2^2 + x2, x3^2 + x3, x4^2 + x4, x5^2 + x5, \
                         x6^2 + x6, x7^2 + x7, x8^2 + x8, x9^2 + x9, x10^2 + x10, \
                         x11^2 + x11, x12^2 + x12, x13^2 + x13, x14^2 + x14, x15^2 + x15, \
                         x16^2 + x16, x17^2 + x17, x18^2 + x18, x19^2 + x19, x20^2 + x20, \
                         x21^2 + x21, x22^2 + x22, x23^2 + x23, x24^2 + x24, x25^2 + x25, \
                         x26^2 + x26, x27^2 + x27, x28^2 + x28, x29^2 + x29, x30^2 + x30])
        sage: I.basis_is_groebner()
        True
        sage: for V in I.variety(): print V  # long time (6s on sage.math, 2011)
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 0, x19: 0, x18: 1, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 0, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 1, x19: 0, x18: 1, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 0, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 0, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 0, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 0, x19: 0, x18: 0, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 1, x4: 1, x19: 0, x18: 0, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 1, x25: 1, x9: 0, x8: 0, x20: 0, x17: 1, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 0, x19: 0, x18: 1, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 0, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}
        {x14: 0, x24: 0, x16: 1, x1: 1, x3: 1, x2: 0, x5: 0, x4: 1, x19: 0, x18: 1, x7: 1, x6: 0, x10: 1, x30: 0, x28: 1, x29: 1, x13: 0, x27: 1, x11: 0, x25: 1, x9: 0, x8: 0, x20: 0, x17: 0, x23: 0, x26: 0, x15: 0, x21: 1, x12: 0, x22: 0}

    Check that the issue at trac ticket #7425 is fixed:

        sage: R.<x, y, z> = QQ[]
        sage: I = R.ideal([x^2-y^3*z, x+y*z])
        sage: I.dimension()
        1
        sage: I.variety()
        Traceback (most recent call last):
        ...
        ValueError: The dimension of the ideal is 1, but it should be 0

    Check that the issue at trac ticket #7425 is fixed:

        sage: S.<t>=PolynomialRing(QQ)
        sage: F.<q>=QQ.extension(t^4+1)
        sage: R.<x,y>=PolynomialRing(F)
        sage: I=R.ideal(x,y^4+1)
        sage: I.variety()
        [...{y: -q^3, x: 0}...]

    Check that computing the variety of the `[1]` ideal is allowed (trac ticket #13977):

        sage: R.<x,y> = QQ[]
        sage: I = R.ideal(1)
        sage: I.variety()
        []

    Check that the issue at trac ticket #16485 is fixed:

        sage: R.<a,b,c> = PolynomialRing(QQ, order='lex')
        sage: I = R.ideal(c^2-2, b-c, a)
        sage: I.variety(QQbar)
        [...a: 0...]

    ALGORITHM:

    Uses triangular decomposition.

  `vector_space_dimension`()

    Return the vector space dimension of the ring modulo this ideal. If the ideal is not zero-dimensional, a TypeError is raised.

    ALGORITHM:

    Uses Singular.

    EXAMPLE:

        sage: R.<u,v> = PolynomialRing(QQ)
        sage: g = u^4 + v^4 + u^3 + v^3
        sage: I = ideal(g) + ideal(g.gradient())
        sage: I.dimension()
        0
        sage: I.vector_space_dimension()
        4

    When the ideal is not zero-dimensional, we return infinity:

        sage: R.<x,y> = PolynomialRing(QQ)
        sage: I = R.ideal(x)
        sage: I.dimension()
        1
        sage: I.vector_space_dimension()
        +Infinity

class `sage.rings.polynomial.multi_polynomial_ideal.``MagmaDefaultContext`(magma=Magma)

  Context to force preservation of verbosity options for Magma’s Groebner basis computation.

class `sage.rings.polynomial.multi_polynomial_ideal.``NCPolynomialIdeal`(ring, gens, coerce=True, side='left')

  Bases: `sage.rings.polynomial.multi_polynomial_ideal.MPolynomialIdeal_singular_repr`, `sage.rings.noncommutative_ideals.Ideal_nc`

  Creates a non-commutative polynomial ideal.

  INPUT:

  - `ring` - the g-algebra to which this ideal belongs
  - `gens` - the generators of this ideal
  - `coerce` (optional - default True) - generators are coerced into the ring before creating the ideal
  - `side` - optional string, either “left” (default) or “twosided”; defines whether this ideal is left of two-sided.

  EXAMPLES:

      sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
      sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
      sage: H.inject_variables()
      Defining x, y, z
      sage: I = H.ideal([y^2, x^2, z^2-H.one_element()],coerce=False) # indirect doctest
      sage: I
      Left Ideal (y^2, x^2, z^2 - 1) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
      sage: H.ideal([y^2, x^2, z^2-H.one_element()], side="twosided")
      Twosided Ideal (y^2, x^2, z^2 - 1) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
      sage: H.ideal([y^2, x^2, z^2-H.one_element()], side="right")
      Traceback (most recent call last):
      ...
      ValueError: Only left and two-sided ideals are allowed.

  `reduce`(p)

    Reduce an element modulo a Groebner basis for this ideal.

    It returns 0 if and only if the element is in this ideal. In any case, this reduction is unique up to monomial orders.

    NOTE:

    There are left and two-sided ideals. Hence,

    EXAMPLE:

        sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
        sage: H.<x,y,z> = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
        sage: I = H.ideal([y^2, x^2, z^2-H.one_element()],coerce=False, side='twosided')
        sage: Q = H.quotient(I); Q
        Quotient of Noncommutative Multivariate Polynomial Ring in x, y, z
         over Rational Field, nc-relations: {z*x: x*z + 2*x,
         z*y: y*z - 2*y, y*x: x*y - z} by the ideal (y^2, x^2, z^2 - 1)
        sage: Q.2^2 == Q.one_element()   # indirect doctest
        True

    Here, we see that the relation that we just found in the quotient is actually a consequence of the given relations:

        sage: I.std()
        Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2)
        of Noncommutative Multivariate Polynomial Ring in x, y, z over
        Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}

    Here is the corresponding direct test:

        sage: I.reduce(z^2)
        1

  `res`(length)

    Computes the resoltuion up to a given length of the ideal.

    NOTE:

    Only left syzygies can be computed. So, even if the ideal is two-sided, then the resolution is only one-sided. In that case, a warning is printed.

    EXAMPLE:

        sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
        sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
        sage: H.inject_variables()
        Defining x, y, z
        sage: I = H.ideal([y^2, x^2, z^2-H.one_element()],coerce=False)
        sage: I.res(3)
        <Resolution>

  `std`()

    Computes a GB of the ideal. It is two-sided if and only if the ideal is two-sided.

    EXAMPLES:

        sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
        sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
        sage: H.inject_variables()
        Defining x, y, z
        sage: I = H.ideal([y^2, x^2, z^2-H.one_element()],coerce=False)
        sage: I.std()
        Left Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}

    If the ideal is a left ideal, then std returns a left Groebner basis. But if it is a two-sided ideal, then the output of std and `twostd()` coincide:

        sage: JL = H.ideal([x^3, y^3, z^3 - 4*z])
        sage: JL
        Left Ideal (x^3, y^3, z^3 - 4*z) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
        sage: JL.std()
        Left Ideal (z^3 - 4*z, y*z^2 - 2*y*z, x*z^2 + 2*x*z, 2*x*y*z - z^2 - 2*z, y^3, x^3) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
        sage: JT = H.ideal([x^3, y^3, z^3 - 4*z], side='twosided')
        sage: JT
        Twosided Ideal (x^3, y^3, z^3 - 4*z) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
        sage: JT.std()
        Twosided Ideal (z^3 - 4*z, y*z^2 - 2*y*z, x*z^2 + 2*x*z, y^2*z - 2*y^2, 2*x*y*z - z^2 - 2*z, x^2*z + 2*x^2, y^3, x*y^2 - y*z, x^2*y - x*z - 2*x, x^3) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {z*x: x*z + 2*x, z*y: y*z - 2*y, y*x: x*y - z}
        sage: JT.std() == JL.twostd()
        True

    ALGORITHM: Uses Singular’s std command

  `syzygy_module`()

    Computes the first syzygy (i.e., the module of relations of the given generators) of the ideal.

    NOTE:

    Only left syzygies can be computed. So, even if the ideal is two-sided, then the syzygies are only one-sided. In that case, a warning is printed.

    EXAMPLE:

        sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
        sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
        sage: H.inject_variables()
        Defining x, y, z
        sage: I = H.ideal([y^2, x^2, z^2-H.one_element()],coerce=False)
        sage: G = vector(I.gens()); G
        d...: UserWarning: You are constructing a free module
        over a noncommutative ring. Sage does not have a concept
        of left/right and both sided modules, so be careful.
        It's also not guaranteed that all multiplications are
        done from the right side.
        d...: UserWarning: You are constructing a free module
        over a noncommutative ring. Sage does not have a concept
        of left/right and both sided modules, so be careful.
        It's also not guaranteed that all multiplications are
        done from the right side.
        (y^2, x^2, z^2 - 1)
        sage: M = I.syzygy_module(); M
        [                                                                         -z^2 - 8*z - 15                                                                                        0                                                                                      y^2]
        [                                                                                       0                                                                          -z^2 + 8*z - 15                                                                                      x^2]
        [                                                              x^2*z^2 + 8*x^2*z + 15*x^2                                                              -y^2*z^2 + 8*y^2*z - 15*y^2                                                                   -4*x*y*z + 2*z^2 + 2*z]
        [                 x^2*y*z^2 + 9*x^2*y*z - 6*x*z^3 + 20*x^2*y - 72*x*z^2 - 282*x*z - 360*x                                                              -y^3*z^2 + 7*y^3*z - 12*y^3                                                                                  6*y*z^2]
        [                                                              x^3*z^2 + 7*x^3*z + 12*x^3                 -x*y^2*z^2 + 9*x*y^2*z - 4*y*z^3 - 20*x*y^2 + 52*y*z^2 - 224*y*z + 320*y                                                                                 -6*x*z^2]
        [  x^2*y^2*z + 4*x^2*y^2 - 8*x*y*z^2 - 48*x*y*z + 12*z^3 - 64*x*y + 108*z^2 + 312*z + 288                                                                           -y^4*z + 4*y^4                                                                                        0]
        [                                                  2*x^3*y*z + 8*x^3*y + 9*x^2*z + 27*x^2                                   -2*x*y^3*z + 8*x*y^3 - 12*y^2*z^2 + 99*y^2*z - 195*y^2                                                                -36*x*y*z + 24*z^2 + 18*z]
        [                                                                           x^4*z + 4*x^4    -x^2*y^2*z + 4*x^2*y^2 - 4*x*y*z^2 + 32*x*y*z - 6*z^3 - 64*x*y + 66*z^2 - 240*z + 288                                                                                        0]
        [x^3*y^2*z + 4*x^3*y^2 + 18*x^2*y*z - 36*x*z^3 + 66*x^2*y - 432*x*z^2 - 1656*x*z - 2052*x                                      -x*y^4*z + 4*x*y^4 - 8*y^3*z^2 + 62*y^3*z - 114*y^3                                                                        48*y*z^2 - 36*y*z]

        sage: M*G
        (0, 0, 0, 0, 0, 0, 0, 0, 0)

    ALGORITHM: Uses Singular’s syz command

  `twostd`()

    Computes a two-sided GB of the ideal (even if it is a left ideal).

    EXAMPLES:

        sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
        sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
        sage: H.inject_variables()
        Defining x, y, z
        sage: I = H.ideal([y^2, x^2, z^2-H.one_element()],coerce=False)
        sage: I.twostd()
        Twosided Ideal (z^2 - 1, y*z - y, x*z + x, y^2, 2*x*y - z - 1, x^2) of Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field...

    ALGORITHM: Uses Singular’s twostd command

class `sage.rings.polynomial.multi_polynomial_ideal.``RequireField`(f)

  Bases: `sage.misc.method_decorator.MethodDecorator`

  Decorator which throws an exception if a computation over a coefficient ring which is not a field is attempted.

  Note

  This decorator is used automatically internally so the user does not need to use it manually.

class `sage.rings.polynomial.multi_polynomial_ideal.``SingularDefaultContext`(singular=Singular)

  Within this context all Singular Groebner basis calculations are reduced automatically.

  AUTHORS:

  - Martin Albrecht
  - Simon King

`sage.rings.polynomial.multi_polynomial_ideal.``is_MPolynomialIdeal`(x)

  Return `True` if the provided argument `x` is an ideal in the multivariate polynomial ring.

  INPUT:

  - `x` - an arbitrary object

  EXAMPLES:

      sage: from sage.rings.polynomial.multi_polynomial_ideal import is_MPolynomialIdeal
      sage: P.<x,y,z> = PolynomialRing(QQ)
      sage: I = [x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y]

  Sage distinguishes between a list of generators for an ideal and the ideal itself. This distinction is inconsistent with Singular but matches Magma’s behavior.

      sage: is_MPolynomialIdeal(I)
      False

      sage: I = Ideal(I)
      sage: is_MPolynomialIdeal(I)
      True

`sage.rings.polynomial.multi_polynomial_ideal.``libsingular_standard_options`(func)

  Decorator to force a reduced Singular groebner basis.

  TESTS:

      sage: P.<a,b,c,d,e> = PolynomialRing(GF(127))
      sage: J = sage.rings.ideal.Cyclic(P).homogenize()
      sage: from sage.misc.sageinspect import sage_getsource
      sage: "buchberger" in sage_getsource(J.interreduced_basis)
      True

  The following tests against a bug that was fixed in trac ticket #11298:

      sage: from sage.misc.sageinspect import sage_getsourcelines, sage_getargspec
      sage: P.<x,y> = QQ[]
      sage: I = P*[x,y]
      sage: sage_getargspec(I.interreduced_basis)
      ArgSpec(args=['self'], varargs=None, keywords=None, defaults=None)
      sage: sage_getsourcelines(I.interreduced_basis)
      (['    @singular_standard_options\n',
        '    @libsingular_standard_options\n',
        '    def interreduced_basis(self):\n',
        ...
        '        return ret\n'], ...)

  Note

  This decorator is used automatically internally so the user does not need to use it manually.

`sage.rings.polynomial.multi_polynomial_ideal.``magma_standard_options`(func)

  Decorator to force default options for Magma.

  EXAMPLE:

      sage: P.<a,b,c,d,e> = PolynomialRing(GF(127))
      sage: J = sage.rings.ideal.Cyclic(P).homogenize()
      sage: from sage.misc.sageinspect import sage_getsource
      sage: "mself" in sage_getsource(J._groebner_basis_magma)
      True

`sage.rings.polynomial.multi_polynomial_ideal.``require_field`

  alias of `RequireField`

`sage.rings.polynomial.multi_polynomial_ideal.``singular_standard_options`(func)

  Decorator to force a reduced Singular groebner basis.

  TESTS:

      sage: P.<a,b,c,d,e> = PolynomialRing(GF(127))
      sage: J = sage.rings.ideal.Cyclic(P).homogenize()
      sage: from sage.misc.sageinspect import sage_getsource
      sage: "buchberger" in sage_getsource(J.interreduced_basis) #indirect doctest
      True

  The following tests against a bug that was fixed in trac ticket #11298:

      sage: from sage.misc.sageinspect import sage_getsourcelines, sage_getargspec
      sage: P.<x,y> = QQ[]
      sage: I = P*[x,y]
      sage: sage_getargspec(I.interreduced_basis)
      ArgSpec(args=['self'], varargs=None, keywords=None, defaults=None)
      sage: sage_getsourcelines(I.interreduced_basis)
      (['    @singular_standard_options\n',
        '    @libsingular_standard_options\n',
        '    def interreduced_basis(self):\n',
        ...
        '        return ret\n'], ...)

  Note

  This decorator is used automatically internally so the user does not need to use it manually.

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