Coverage for local/lib/python2.7/site-packages/sage/quadratic_forms/constructions.py : 67%

""" 

Some Extras 

""" 

## 

## Some extra routines to make the QuadraticForm class more useful. 

## 

from sage.rings.all import ZZ 

from sage.rings.polynomial.polynomial_element import is_Polynomial 

from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing 

from sage.quadratic_forms.quadratic_form import QuadraticForm 

def BezoutianQuadraticForm(f, g): 

r""" 

Compute the Bezoutian of two polynomials defined over a common base ring. This is defined by 

.. MATH:: 

{\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 

""" 

## Check that f and g are polynomials with a common base ring 

if not is_Polynomial(f) or not is_Polynomial(g): 

raise TypeError("Oops! One of your inputs is not a polynomial. =(") 

if f.base_ring() != g.base_ring(): ## TO DO: Change this to allow coercion! 

raise TypeError("Oops! These polynomials are not defined over the same coefficient ring.") 

## Initialize the quadratic form 

R = f.base_ring() 

P = PolynomialRing(R, ['x','y']) 

a, b = P.gens() 

n = max(f.degree(), g.degree()) 

Q = QuadraticForm(R, n) 

## Set the coefficients of Bezoutian 

bez_poly = (f(a) * g(b) - f(b) * g(a)) // (b - a) ## Truncated (exact) division here 

for i in range(n): 

for j in range(i, n): 

if i == j: 

Q[i,j] = bez_poly.coefficient({a:i,b:j}) 

else: 

Q[i,j] = bez_poly.coefficient({a:i,b:j}) * 2 

return Q 

def 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 ] 

""" 

r = ZZ(r) 

## Check that the multiplicity is a natural number 

if r < 1: 

raise TypeError("The multiplicity r must be a natural number.") 

H = QuadraticForm(R, 2, [0, 1, 0]) 

return sum([H for i in range(r-1)], H)