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Quadratic Spaces

A quadratic form on a vector space V over a field F is a function Q : V to F such that

(i)
Q(av) = a2 Q(v) for all a∈F and all v∈V; and

(ii)
the function β: V x V to F defined by β(u, v) := Q(u + v) - Q(u) - Q(V) is a bilinear form on V (called the polar form of Q).

A vector space V with an attached quadratic form is called a quadratic space. In Magma the polar form of a quadratic space is the Gram matrix B of the space. If the characteristic of the field is not 2, the value of the quadratic form on a row vector v is (12)/(v * B * vT).

A vector v≠0 is isotropic if β(v, v) = 0; it is defined to be singular if Q(v) = 0.

The radical of V is the subspace of vectors u such that β(u, v) = 0 for all v∈V.

QuadraticSpace(Q) : AlgMatElt -> ModTupFld
The quadratic space defined by the quadratic form represented by a square matrix Q over a field.
QuadraticNorm(v) : ModTupFldElt -> FldElt
The value Q(v), where Q is the quadratic form attached to the ambient space of the vector v.
IsSingular(V) : ModTupFld -> BoolElt, ModTupFldElt
Determine whether the quadratic space V contains a singular vector; if it does, assign a representative as the second return value.
HyperbolicPair(V,u) : ModTupFld, ModTupFldElt -> ModTupFldElt
If u is a singular vector in a quadratic space V this function returns a singular vector v such that β(u, v) = 1, where βis the polar form of the space, provided u is not in the radical of V.
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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012