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                                       # needs sage.libs.singular
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 -12 ]
[ * -15 ]

>>> from sage.all import *
>>> R = PolynomialRing(ZZ, 'x')
>>> f = R([Integer(1),Integer(2),Integer(3)])
>>> g = R([Integer(2),Integer(5)])
>>> Q = BezoutianQuadraticForm(f, g); Q                                       # needs sage.libs.singular
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 -12 ]
[ * -15 ]


AUTHORS:

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

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 ]

>>> from sage.all import *
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 0 1 ]
[ * 0 ]