(FUNDAMENTAL AND APPLIED MATHEMATICS)

2000, VOLUME 6, NUMBER 3, PAGES 649-668

## Exponential Diophantine equations in rings of positive characteristic

A. Ya. Belov
A. A. Chilikov

Abstract

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In this work we prove the algorithmical solvability of the exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations


 s S i = 1 Pij(n1,¼,nt) bij0aij1n1bij1¼aijtntbijt = 0,

where $b$ijk,aijk are constants from matrix ring of characteristic $p$, $n$i are indeterminates. For any solution á n1, ¼ ,nt ñ of the system we construct the word (over alphabet which contains $pt$ symbols) `a0¼`aq, where `ai is a $t$-tuple á n1(i), ¼ ,nt(i) ñ, $n\left(i\right)$ is the $i$-th digit in the $p$-adic representation of $n$. The main result of this work is: the set of words, corresponding in this sense to the solutions of the system of exponential-Diophantine equations is a regular language (i. e. recognizible by a finite automaton). There is an effective algorithm which calculates this language.

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