anisotropicDimension -- returns the anisotropic dimension of a symmetric bilinear form

Synopsis

• Usage:

  anisotropicDimension(beta)

• Inputs:
  • beta, a Grothendieck Witt Class, over a field $k$ where $k$ is the complex numbers, reals, rationals, or a finite field
• Outputs:
  • an integer, the rank of the anisotropic part of $\beta$

Description

By Witt decomposition, any form decomposes uniquely as $\beta \cong k \mathbb{H} \oplus \beta_a$ where the form $\beta_a$ is anisotropic. The rank of $\beta_a$ is called the anisotropic dimension of $\beta$.

The anisotropic dimension of a form defined over the rationals is the maximum of the anistropic dimension at each of the completions of $\mathbb{Q}$.

See also

• WittIndex -- returns the Witt index of a symmetric bilinear form
• anisotropicDimensionQp -- returns the anisotropic dimension of a rational symmetric bilinear form over the p-adics
• anisotropicDimension -- returns the anisotropic dimension of a symmetric bilinear form