On Diophantine exponents for Laurent series over a finite field

TL;DR

This paper investigates Diophantine exponents of Laurent series over finite fields, constructing sequences with prescribed exponents and providing examples where certain exponents differ, advancing understanding of their properties.

Contribution

It establishes the existence of Laurent series with specific Diophantine exponents and provides explicit examples illustrating differences between related exponents.

Findings

Existence of Laurent series with prescribed exponents for w_n and w_n^*

Construction of sequences with constant Diophantine exponents

Explicit examples where w_n and w_n^* differ for n ≥ 2

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 $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j)_{j\geq 1}$ and a sequence of algebraic Laurent series $(\xi_j)_{j\geq 1}$ such that deg $\xi_j=p^{k_j}+1$ and \begin{equation} w_1(\xi_j)=w_1 ^{*}(\xi_j)=\ldots =w_n(\xi_j)=w_n ^{*}(\xi_j)=w \end{equation} for any $j\geq 1$. For each $n\geq 2$, we give explicit examples of Laurent series $\xi $ for which $w_n(\xi )$ and $w_n^{*}(\xi )$ are different.