arXiv subjects

Jun-ichi Matsushita

Publications and source records attributed to Jun-ichi Matsushita.

2 recordsLinked to original sources

arXiv2024

Dual instability measures of a subspace of $P^{n}(K)$ under a subgroup of $\operatorname{Aut}(K)$

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 $k$.

math.NT↗

arXiv2011

An algebraic result on the topological closure of the set of rational points on a sphere whose center is non-rational, II

Let S be a sphere in R^n such that S\capQ^n\neq\emptyset and let Cl denote the closure operator in the Euclidean topology of R^n. If the center of S is in Q^n, then Cl(S\capQ^n) is S, as is easily proved. If the center of S is not in Q^n, then what is Cl(S\capQ^n)? This question, which was answered partially in the author's paper [Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 7, 146--149], is answered completely in this paper by representing Cl(S\capQ^n) in terms of the group of Q-automorphisms of the algebraic closure of Q(γ_1,...,γ_n) in C, where γ_1,...,γ_n denote the coordinates of the center of S.

math.NT↗