Diophantine approximation in positive characteristic
type: Item
instance of: scholarly article
title: Diophantine approximation in positive characteristic
zbMATH Open document ID: 0661.10050
DOI: 10.1007/BF01850290
published in: Periodica Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: Let K be a field of characteristic \(p>0\). A classical order is defined on Kx as follows: let \(f(x)=\sum^{\infty}_{n=t}a_ nx^ n\) with \(a_ t\neq 0\). Then one puts \(ord(f)=t\). Mahler then defind \(\alpha (y)=\limsup_{H(r)\to \infty,\quad r\in K(x)}(ord(y-t)/H(r))\) with \(H(P/Q)=\max (\deg P, \deg Q)\). Also for y algebraic over K(x), [K(x,y): K(X)] is denoted by d(y). Mahler proved that \(\alpha\) (y)\(\leq d(y).\) Here the author considers \(y\in Kx\) in the form \((ay^ q+b)/(cy^ q+d)\) with a, b, c, \(d\in Kx\), ad-bc\(\neq 0\) and with q a power of p. The author wants to study the approximation of y by rational functions (with respect to the valuation ord). The main theorem shows that, provided \(d(y)>1\), for any \(r\in K(x),\) either \(H(r)\leq A(q-1),\) or ord(y-1)\(\leq \alpha (y) H(r)+\frac{\alpha (y)A}{q-1}+\frac{B+2 ord(y)}{q-1}\) with \(A=\max (\deg a\), deg b, deg c, deg d) and \(B=ord(ad-bc)\). Applications and examples are given. They show the main theorem is the best possible.
reviewed by: Alain Escassut
Mathematics Subject Classification ID: 11J99
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 4079546
zbMATH Keywords: rational functions
zbMATH Keywords: positive characteristic
zbMATH Keywords: order
author: José Felipe Voloch
MaRDI profile type: Publication
cites work: Q3663376
cites work: On a Theorem of Liouville in Fields of Positive Characteristic
cites work: Q4055730
full work available at URL: https://doi.org/10.1007/bf01850290
OpenAlex ID: W2076744257
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description: scientific article; zbMATH DE number 4079546