On Diophantine exponents for Laurent series over a finite field

In this paper, we study the properties of Diophantine exponents wn and wn⁎ for Laurent series over a finite field. We prove that for an integer n≥1 and a rational number w>2n−1, there exist a strictly increasing sequence of positive integers (kj)j≥1 and a sequence of algebraic Laurent series (ξj)j≥1 such that deg⁡ξj=pkj+1 andw1(ξj)=w1⁎(ξj)=…=wn(ξj)=wn⁎(ξj)=w for any j≥1. For each n≥2, we give explicit examples of Laurent series ξ for which wn(ξ) and wn⁎(ξ) are different.