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getSumDecomposition -- produces a simplified diagonal representative of a Grothendieck-Witt class

Description

Given a symmetric bilinear form beta over $\mathbb{Q},$ $ \mathbb{R},$ $\mathbb{C}$ or a finite field of characteristic not 2, we decompose it as a sum of some number of hyperbolic and rank one forms.

i1 : M = matrix(RR, {{2.091,2.728,6.747},{2.728,7.329,6.257},{6.747,6.257,0.294}});

                3         3
o1 : Matrix RR    <-- RR
              53        53
i2 : beta = makeGWClass M;
i3 : getSumDecomposition beta
../../../../../Macaulay2/packages/A1BrouwerDegrees/Code/GWInvariants.m2:25:44:(2):[34]: error: Matrix is not symmetric
../../../../../Macaulay2/packages/A1BrouwerDegrees/Code/AnisotropicDimension.m2:108:38:(2):[33]: --back trace--
../../../../../Macaulay2/packages/A1BrouwerDegrees/Code/AnisotropicDimension.m2:131:27:(2):[32]: --back trace--
../../../../../Macaulay2/packages/A1BrouwerDegrees/Code/AnisotropicDimension.m2:139:49:(2):[31]: --back trace--
../../../../../Macaulay2/packages/A1BrouwerDegrees/Code/Decomposition.m2:259:21:(2):[30]: --back trace--
../../../../../Macaulay2/packages/A1BrouwerDegrees/Code/Decomposition.m2:281:61:(2):[29]: --back trace--
currentString:3:19:(3):[28]: --back trace--

Over $\mathbb{R}$ there are only two square classes and a form is determined uniquely by its rank and signature [L05, II Proposition 3.2]. A form defined by the $3\times 3$ Gram matrix M above is isomorphic to the form $\langle 1,-1,1\rangle $.

i4 : -- example results terminated prematurely
i5 : -- example results terminated prematurely
i6 : -- example results terminated prematurely

Over $\mathbb{F}_{q}$ forms can similarly be diagonalized, in the above case as $\langle 1,-1,1,-6 \rangle$.

Citations:

See also

Ways to use getSumDecomposition:

  • getSumDecomposition(GrothendieckWittClass)

For the programmer

The object getSumDecomposition is a method function.


The source of this document is in A1BrouwerDegrees/Documentation/DecompositionDoc.m2:32:0.