anisotropicPart -- returns the anisotropic part of a Grothendieck Witt class

Synopsis

• Usage:

  anisotropicPart(beta)

• Inputs:
  • beta, a Grothendieck Witt Class, a symmetric bilinear form defined over a field $k$
• Outputs:
  • a Grothendieck Witt Class, the anisotropic part of $\beta$

Description

Given a form $\beta$ we may compute its anisotropic part inductively by reference to its anisotropic dimension. Over the complex numbers and the reals this is trivial, and over finite fields it is a fairly routine computation, however over the rationals some more sophisticated algorithms are needed from the literature. For this methods we implement algorithms developed for number fields by Koprowski and Rothkegel [KR23]. Note also that a Chinese Remainder Theorem method is needed in reducing from anisotropic dimension three as in [KR23, Algorithm 7], so we import one from the Parametrization package.

i1 : alpha = diagonalForm(QQ,(3,-3,2,5,1,-9));

i2 : anisotropicPart(alpha)

o2 = GrothendieckWittClass{cache => CacheTable{}}
                           matrix => | 2 0  |
                                     | 0 20 |

o2 : GrothendieckWittClass

Citations:

• [KR23] P. Koprowski and B. Rothkegel, The anisotropic part of a quadratic form over a number field, Journal of Symbolic Computatoin, 2023.

See also

• anisotropicDimension -- returns the anisotropic dimension of a symmetric bilinear form
• WittIndex -- returns the Witt index of a symmetric bilinear form