Exponential Diophantine equations in rings of positive characteristic

A. Ya. Belov ; A. A. Chilikov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 3, pp. 649-668.

Résumé

In this work we prove the algorithmical solvability of the exponential-Diophan-tine 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.