Spaces with Vanishing Characteristic Coefficients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917618098634752 |
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| author | Chapman, Adam |
| author_facet | Chapman, Adam |
| contents | 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 $kv+(p-1-k)w=u$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_13080 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spaces with Vanishing Characteristic Coefficients Chapman, Adam Rings and Algebras Number Theory primary 16K20, secondary 16W60, 11B50, 12E15 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 $kv+(p-1-k)w=u$. |
| title | Spaces with Vanishing Characteristic Coefficients |
| topic | Rings and Algebras Number Theory primary 16K20, secondary 16W60, 11B50, 12E15 |
| url | https://arxiv.org/abs/2403.13080 |