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

Synopsis

• Usage:

  sumDecomposition(beta)

• Inputs:
  • beta, a Grothendieck Witt Class, a symmetric bilinear form defined over a field $k$
• Outputs:
  • a Grothendieck Witt Class, 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.

See also

• sumDecompositionString -- produces a simplified diagonal representative of a Grothendieck Witt class