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

Usage:

getAnisotropicPart(beta)
Inputs:
- beta, a Grothendieck-Witt Class, over $\mathbb{C},\mathbb{Q},\mathbb{R}$, or a finite field of characteristic not two
Outputs:
- a Grothendieck-Witt Class, the anisotropic part of the Grothendieck-Witt class

Description

By the Witt Decomposition theorem, any Grothendieck-Witt class can be decomposed uniquely into a sum of hyperbolic forms and an anisotropic part $\beta\cong\beta_{a}\oplus n\mathbb{H}$ where the form $\beta_{a}$ is anisotropic. This method returns the anisotropic part $\beta_{a}$ of the Grothendieck-Witt class $\beta$.

Over the complex and real numbers, this is straightforward, and over finite fields it is a fairly routine computation. Over the rational number, 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

References

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

Ways to use getAnisotropicPart:

getAnisotropicPart(GrothendieckWittClass)
getAnisotropicPart(Matrix)

For the programmer

The object getAnisotropicPart is a method function.

The source of this document is in A1BrouwerDegrees/Documentation/DecompositionDoc.m2:128:0.

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

Description

References

See also

Ways to use getAnisotropicPart:

For the programmer