FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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

where $b$_ijk,a_ijk are constants from matrix ring of characteristic $p$, $n$_i are indeterminates. For any solution $$á n₁, ¼ ,n_t ñ of the system we construct the word (over alphabet which contains $pt$ symbols) $$`a<sub>0</sub>¼`a_q, where $$`a_i is a $t$-tuple $$á n₁⁽ⁱ⁾, ¼ ,n_t⁽ⁱ⁾ ñ, $n(i)$ 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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Last modified: December 8, 2000