next | previous | forward | backward | up | index | toc

# sumDecomposition -- produces a simplified diagonal representative of a Grothendieck Witt class

## Synopsis

• Usage:
sumDecomposition(beta)
• Inputs:
• beta, , a symmetric bilinear form defined over a field $k$
• Outputs:
• , a diagonal representative of the Grothendieck Witt class of the input form

## Description

Given a symmetric bilinear form beta over a field $k$, 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 = gwClass(M); i3 : sumDecomposition(beta) o3 = GrothendieckWittClass{cache => CacheTable{}} matrix => | 1 0 0 | | 0 1 0 | | 0 0 -1 | o3 : GrothendieckWittClass

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 : M = matrix(GF(13),{{9,1,7,4},{1,10,3,2},{7,3,6,7},{4,2,7,5}}); 4 4 o4 : Matrix (GF 13) <-- (GF 13) i5 : beta = gwClass(M); i6 : sumDecomposition(beta) o6 = GrothendieckWittClass{cache => CacheTable{} } matrix => | 1 0 0 0 | | 0 -5 0 0 | | 0 0 1 0 | | 0 0 0 -1 | o6 : GrothendieckWittClass

Citations:

• [L05] T.Y. Lam, Introduction to quadratic forms over fields, American Mathematical Society, 2005.