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 RodriguezVillegas, 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 ]