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

Description

By the Witt Decomposition Theorem, any non-degenerate form decomposes uniquely as $\beta \cong n \mathbb{H} \oplus \beta_a$ where the form $\beta_a$ is anisotropic. We compute the anisotropic part $\beta_a$ inductively by reference to its anisotropic dimension. Over the complex numbers and real numbers this is straightforward, and over finite fields it is a fairly routine computation. Over the rational numbers some more sophisticated algorithms are needed from the literature. For this we implement algorithms developed for number fields by Koprowski and Rothkegel [KR23].

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

i2 : getAnisotropicPart alpha

o2 = | 2 0 |
     | 0 5 |

o2 : GrothendieckWittClass

Citations:

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

See also

• getAnisotropicDimension -- returns the anisotropic dimension of a symmetric bilinear form
• getWittIndex -- returns the Witt index of a symmetric bilinear form
• getSumDecomposition -- produces a simplified diagonal representative of a Grothendieck-Witt class
• getSumDecompositionString -- produces a simplified diagonal representative of a Grothendieck-Witt class