Dual instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2016
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| _version_ | 1866929234765676544 |
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| author | Matsushita, Jun-ichi |
| author_facet | Matsushita, Jun-ichi |
| contents | Let $K$ be a commutative field and let $V$ be a subspace of $P^{n}(K)$. Let $Γ$ be a subgroup of $\operatorname{Aut}(K)$ and let $Γ$ act on $P^{n}(K)$ by $σ((x_{i})_{0\leq i\leq n})=(σ(x_{i}))_{0\leq i\leq n}$ for $σ\inΓ$ and $(x_{i})_{0\leq i\leq n}\in P^{n}(K)$. In this paper, we ask `how much' unstable $V$ is under $Γ$ by asking how much higher (or lower) dimension the join (or the meet) of $σ(V)$ ($σ\inΓ$) has than $V$, and answer it in terms of the Plücker coordinates of $V$ and the invariant field $k$ of $Γ$, through presenting dual `irrationality' measures of $V$ over $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1603_01349 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Dual instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$ Matsushita, Jun-ichi Number Theory Let $K$ be a commutative field and let $V$ be a subspace of $P^{n}(K)$. Let $Γ$ be a subgroup of $\operatorname{Aut}(K)$ and let $Γ$ act on $P^{n}(K)$ by $σ((x_{i})_{0\leq i\leq n})=(σ(x_{i}))_{0\leq i\leq n}$ for $σ\inΓ$ and $(x_{i})_{0\leq i\leq n}\in P^{n}(K)$. In this paper, we ask `how much' unstable $V$ is under $Γ$ by asking how much higher (or lower) dimension the join (or the meet) of $σ(V)$ ($σ\inΓ$) has than $V$, and answer it in terms of the Plücker coordinates of $V$ and the invariant field $k$ of $Γ$, through presenting dual `irrationality' measures of $V$ over $k$. |
| title | Dual instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/1603.01349 |