Constructions of quadratic forms¶
- sage.quadratic_forms.constructions.BezoutianQuadraticForm(f, g)[source]¶
- 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 
 
- sage.quadratic_forms.constructions.HyperbolicPlane_quadratic_form(R, r=1)[source]¶
- Construct 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 * >>> HyperbolicPlane_quadratic_form(ZZ) Quadratic form in 2 variables over Integer Ring with coefficients: [ 0 1 ] [ * 0 ]