FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

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
```

$$

\sum_{i=1}^{s} P_{ij}(n_1,\ldots,n_t)
b_{ij0}a_{ij1}^{n_1}b_{ij1}\ldots a_{ijt}^{n_t}b_{ijt}=0

$$

where $b_{ijk},a_{ijk}$ are constants from matrix ring of
characteristic $p$ , $n_i$ are indeterminates. For any solution
$\langle n_1,\ldots,n_t \rangle$ of the system we construct the word
(over alphabet which contains $p^t$ symbols)
$\overline\alpha_0 \ldots \overline\alpha_q$ , where
$\overline\alpha_i$ is a $t$ -tuple
$\langle n_1^{(i)},\ldots,n_t^{(i)} \rangle$ ,
$n^{(i)}$ 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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