hyperbolicForm -- the Grothendieck-Witt class of a hyperbolic form

Synopsis

• Usage:

  hyperbolicForm(k)

  hyperbolicForm(k,n)

• Inputs:
  • k, a ring, a field
  • n, an integer, an even number, giving an optional rank $n$ for a totally hyperbolic form
• Outputs:
  • a Grothendieck Witt Class, the hyperbolic form $\mathbb{H} = \langle 1, -1\rangle \in \text{GW}(k)$ or the totally hyperbolic form $\frac{n}{2}\mathbb{H}$ if an optional rank is specified

Description

By default outputs the rank two hyperbolic form over the input field $k$.

i1 : hyperbolicForm(GF(7))

o1 = GrothendieckWittClass{cache => CacheTable{}}
                           matrix => | 1 0  |
                                     | 0 -1 |

o1 : GrothendieckWittClass

Specifying a rank yields a copy of sums of the rank two hyperbolic form. Only even rank inputs are accepted.

i2 : hyperbolicForm(RR,4)

o2 = GrothendieckWittClass{cache => CacheTable{}  }
                           matrix => | 1 0  0 0  |
                                     | 0 -1 0 0  |
                                     | 0 0  1 0  |
                                     | 0 0  0 -1 |

o2 : GrothendieckWittClass

See also

• isAnisotropic -- determines whether a Grothendieck-Witt class is anisotropic
• sumDecomposition -- produces a simplified diagonal representative of a Grothendieck Witt class
• sumDecompositionString -- produces a simplified diagonal representative of a Grothendieck Witt class