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Spaces with Vanishing Characteristic Coefficients

Adam Chapman

Abstract

We prove that the maximal dimension of a subspace $V$ of the generic tensor product of $m$ symbol algebras of prime degree $p$ with $\operatorname{Tr}(v^{p-1})=0$ for all $v\in V$ is $\frac{p^{2m}-1}{p-1}$. The same upper bound is thus obtained for $V$ with $\operatorname{Tr}(v)=\operatorname{Tr}(v^2)=\dots=\operatorname{Tr}(v^{p-1})=0$ for all $v \in V$. We make use of the fact that for any subset $S$ of $\underbrace{\mathbb{F}_p \times \dots \times \mathbb{F}_p}_{n \ \text{times}}$ of $|S| > \frac{p^{n}-1}{p-1}$, for all $u\in V$ there exist $v,w\in S$ and $k\in [\![0,p-1]\!]$ such that $kv+(p-1-k)w=u$.

Spaces with Vanishing Characteristic Coefficients

Abstract

We prove that the maximal dimension of a subspace of the generic tensor product of symbol algebras of prime degree with for all is . The same upper bound is thus obtained for with for all . We make use of the fact that for any subset of of , for all there exist and such that .
Paper Structure (6 sections, 7 theorems, 2 equations)

This paper contains 6 sections, 7 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Lines in $\mathbb{F}_p^{\times n}$
  3. Symbol Algebras
  4. Trace conditions for vanishing characteristic coefficients
  5. Bounding the dimension
  6. The trace of the square and open questions

Key Result

Lemma 2.1

Given a prime $p$, $n \in \mathbb{N}$ and a set $S \in \mathbb{F}_p^{\times n}$ of $|S|>\frac{p^n-1}{p-1}$, every point $P \in V$ lives on a line passing through two distinct points of $S$.

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Theorem 5.1
  • proof
  • Remark 5.2
  • ...and 8 more