1. On an Extension of Hoffmann's Separation Theorem for Quadratic Forms.
- Author
-
Scully, Stephen
- Subjects
- *
KERNEL functions , *QUADRATIC forms , *QUADRICS , *LOGICAL prediction , *INTEGERS , *CHAR - Abstract
Let |$p$| and |$q$| be anisotropic non-degenerate quadratic forms of dimension |$\geq 2$| over an arbitrary field |$F$| , let |$s$| be the unique non-negative integer for which |$2^s<{\textrm{dim}(p)} \leq 2^{s+1}$| , and let |$k$| be the dimension of the anisotropic part of |$q$| after extension to |$F(p)$|. A recent conjecture of the author then asserts that |${\textrm{dim}(q)}$| must lie within |$k$| of an integer multiple of |$2^{s+1}$|. This statement, which holds trivially if |$k \geq 2^s -1$| , represents a natural generalization of the well-known separation theorem of Hoffmann, bridging a gap between it and certain classical results on the Witt kernels of function fields of quadrics. In the present article, we prove the conjecture in the case where |$\textrm{char}(F) \neq 2$| and |${\textrm{dim}(p)}> 2k - 2^{s-1}$|. This implies, in particular, that the conjecture holds if |$\textrm{char}(F) \neq 2$| and either |$k \leq 2^{s-1} + 2^{s-2}$| or |${\textrm{dim}(p)} \geq 2^s + 2^{s-1} - 4$|. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF