On Diophantine exponents for Laurent series over a finite field

Tomohiro Ooto

doi:10.1016/j.jnt.2017.09.008

paper

On Diophantine exponents for Laurent series over a finite field

arXiv:1611.05719 · doi:10.1016/j.jnt.2017.09.008

Abstract

In this paper, we study properties of the Diophantine exponents $w_{n}$ and $w_{n}^{*}$ for Laurent series over a finite field. We prove that for an integer $n \geq 1$ and a rational number $w > 2 n - 1$ , there exist a strictly increasing sequence of positive integers $(k_{j})_{j \geq 1}$ and a sequence of algebraic Laurent series $(ξ_{j})_{j \geq 1}$ such that deg $ξ_{j} = p^{k_{j}} + 1$ and \begin{equation} w_1(ξ_j)=w_1 ^{*}(ξ_j)=\ldots =w_n(ξ_j)=w_n ^{*}(ξ_j)=w \end{equation} for any $j \geq 1$ . For each $n \geq 2$ , we give explicit examples of Laurent series $ξ$ for which $w_{n} (ξ)$ and $w_{n}^{*} (ξ)$ are different.

22 pages

On Diophantine exponents for Laurent series over a finite field

Abstract

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