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 ]