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# 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:
• , 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

• 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

## Ways to use hyperbolicForm :

• hyperbolicForm(InexactFieldFamily)
• hyperbolicForm(InexactFieldFamily,ZZ)
• hyperbolicForm(Ring)
• hyperbolicForm(Ring,ZZ)

## For the programmer

The object hyperbolicForm is .