Differential equations and Diophantine approximation in positive characteristic
type: Item
instance of: scholarly article
title: Differential equations and Diophantine approximation in positive characteristic
zbMATH Open document ID: 0953.11024
DOI: 10.1007/PL00010080
published in: Monatshefte für Mathematik
publication date: 2001-01-26T00:00:00Z
review text: The subject is diophantine approximation of formal Laurent series \(F=\sum_{n=-m}^\infty f_nX^{-n}\) over some field \(K\); here \(\deg F=\deg_X F=m\). Let \(P\) and \(Q\neq 0\) denote polynomials in \(X\). It is well known that Roth's Theorem fails in positive characteristic and indeed that Liouvile's Theorem is best possible. However the authors remark that they have shown [J. Reine Angew. Math. 473, 195-206 (1996; Zbl 0844.11046)] that Thue's method leads to \(\deg(F-P/Q)> -(\frac 12\lfloor r\rfloor +1 +\varepsilon)\deg Q\), for algebraic elements \(F\) of degree \(r\), provided that \(F\) is not of the shape \(F=(AF^{p^s}+B)/(CF^{p^s}+D)\) for polynomials \(A\), \(B\), \(C\), and \(D\). In the present note the authors first detail a version of \textit{C. F. Osgood}'s argument [Nederl. Akad. Wet, Proc., Ser. A 78, 401 (1975; Zbl 0313.10036)] showing that, if \(F\) does not satisfy a rational Riccati differential equation, the arbitrary \(\varepsilon>0\) may be omitted above (up to an \(O(1)\) correction). They then apply a remark of \textit{J. F. Voloch} [Monatsh. Math. 119, 321-325 (Zbl 0827.11039)] to obtain their main result that if \(K\) is a finite field the \(\varepsilon\) may also be omitted under their condition. In passing one sees that if a rational Riccati equation has no rational solution then a solution \(F\in K((1/X))\) satisfies \(\deg(F-P/Q)> -2\deg Q-O(1)\). The authors note that, in characteristic \(2\), there are rational Riccati equations without a rational solution. This is not so in characteristic \(p=3\). The cases \(p>3\) remain open.
Mathematics Subject Classification ID: 11J61
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 1340540
zbMATH Keywords: diophantine approximation of formal Laurent series
zbMATH Keywords: rational Riccati equation
author: Alain Lasjaunias
author: Bernard de Mathan
MaRDI profile type: Publication
full work available at URL: pl00010080
OpenAlex ID: W2083360194
description: scientific article; zbMATH DE number 1340540