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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