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# Some Extras

`sage.quadratic_forms.constructions.``BezoutianQuadraticForm`(f, g)

  Compute the Bezoutian of two polynomials defined over a common base ring. This is defined by

  \[{\rm Bez}(f, g) := \frac{f(x) g(y) - f(y) g(x)}{y - x}\]

  and has size defined by the maximum of the degrees of \(f\) and \(g\).

  INPUT:

  - \(f\), \(g\) – polynomials in \(R[x]\), for some ring \(R\)

  OUTPUT:

  a quadratic form over \(R\)

  EXAMPLES:

      sage: R = PolynomialRing(ZZ, 'x')
      sage: f = R([1,2,3])
      sage: g = R([2,5])
      sage: Q = BezoutianQuadraticForm(f, g) ; Q
      Quadratic form in 2 variables over Integer Ring with coefficients:
      [ 1 -12 ]
      [ * -15 ]

  AUTHORS:

  - Fernando Rodriguez-Villegas, Jonathan Hanke – added on 11/9/2008

`sage.quadratic_forms.constructions.``HyperbolicPlane_quadratic_form`(R, r=1)

  Constructs the direct sum of \(r\) copies of the quadratic form \(xy\) representing a hyperbolic plane defined over the base ring \(R\).

  INPUT:

  - \(R\): a ring
  - \(n\) (integer, default 1) number of copies

  EXAMPLES:

      sage: HyperbolicPlane_quadratic_form(ZZ)
      Quadratic form in 2 variables over Integer Ring with coefficients:
      [ 0 1 ]
      [ * 0 ]

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