Yann Bugeaud: On simultaneous approximation in fields of power series, 191-202

Abstract:

Let ${\mathbb{F}}_q ((T^{-1}))$ denote the field of power series over the field ${\mathbb{F}}_q$ of $q$ elements, equipped with the absolute value $\vert \cdot \vert$ normalised in such a way that $\vert T\vert = q$. For a power series $\xi$ in ${\mathbb{F}}_q ((T^{-1}))$ and a positive integer $n$, we denote by $\lambda_n (\xi)$ the supremum of the real numbers $\lambda$ for which

$$0 < \max\{ \vert Q(T) \xi - P_1 (T)\vert, \ldots , \vert Q(T) \xi^n - P_n (T)\vert \} < q^{-\lambda \deg(Q)}$$

has infinitely many solutions in polynomials $Q(T), P_1(T), \ldots , P_n(T)$ in ${\mathbb{F}}_q[T]$. We study the set of values taken by the function $\lambda_n$ over the power series in ${\mathbb{F}}_q ((T^{-1}))$ and over the algebraic power series in ${\mathbb{F}}_q ((T^{-1}))$.

Key Words: Diophantine approximation, power series field, simultaneous approximation.

2010 Mathematics Subject Classification: Primary 11J61; Secondary 11J13.