Compute the Bezoutian of two polynomials defined over a common base ring. This is defined by
and has size defined by the maximum of the degrees of \(f\) and \(g\).
INPUT:
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:
Constructs the direct sum of \(r\) copies of the quadratic form \(xy\) representing a hyperbolic plane defined over the base ring \(R\).
INPUT:
EXAMPLES:
sage: HyperbolicPlane_quadratic_form(ZZ)
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 0 1 ]
[ * 0 ]