Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics
Abstract
Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ of characteristic $\neq 2$, let $s$ be the unique nonnegative integer such that $2^s < \mathrm{dim}(p) \leq 2^{s+1}$, and let $k$ denote the dimension of the anisotropic part of $q$ after scalar extension to the function field $F(p)$ of $p$. We conjecture that $\mathrm{dim}(q)$ must lie within $k$ of a multiple of $2^{s+1}$. This can be viewed as a direct generalization of Hoffmann's separation theorem. Among other cases, we prove that the conjecture is true if $k<2^{s1}$. When $k=0$, this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism $W(F)\rightarrow W(F(p))$ has dimension divisible by $2^{s+1}$.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 DOI:
 10.48550/arXiv.1609.07100
 arXiv:
 arXiv:1609.07100
 Bibcode:
 2016arXiv160907100S
 Keywords:

 Mathematics  Commutative Algebra;
 11E04;
 14E05;
 14C15
 EPrint:
 Rewritten