On Diophantine exponents for Laurent series over a finite field

In this paper, we study properties of the Diophantine exponents $w_n$ and $w_n^{\ast }$ for Laurent series over a finite field. We prove that for an integer $n\ge 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j{)}_{j\ge 1}$ and a sequence of algebraic Laurent series $({\xi }_j{)}_{j\ge 1}$ 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 $j\ge 1$. For each $n\ge 2$, we give explicit examples of Laurent series $\xi$ for which $w_n(\xi )$ and $w_n^{\ast }(\xi )$ are different.