Normal basises of algebras and Exponential Diophantine equations in rings of positive characteristic

TL;DR

This paper proves the algorithmic solvability of exponential-Diophantine equations over rings of positive characteristic by translating solutions into regular languages recognized by finite automata.

Contribution

It introduces a novel method to represent solutions of exponential-Diophantine equations as regular languages, enabling effective computation and analysis.

Findings

Solutions correspond to regular languages recognizable by finite automata

An effective algorithm for constructing these languages is provided

The approach applies to rings of positive characteristic and matrix representations

Abstract

In this paper we discourse basises of representable algebras. This question lead to arithmetic problems. We prove algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations $$ \sum\limits_{i=1}^s P_{ij}(n_1,\dots,n_t) b_{ij0} a_{ij1}^{n_1} b_{ij1} \dots a_{ijt}^{n_t}b_{ijt}=0 $$ where $b_{ijk},a_{ijk}$ are constants from matrix ring of characteristic $p$, $n_i$ are indeterminates. For any solution $(n_1,\dots,n_t)$ of the system we construct a word (over an alphabet containing $p^t$ symbols) ${\overline \alpha_0},\dots,{\overline \alpha_q}$ where ${\overline \alpha_i}$ is a $t$-tuple $\langle n_1^{(i)},\dots,n_t^{(i)}\rangle$, $n^{(i)}$ is the $i$-th digit in the $p$-adic representation of $n$. The main result of this paper is as follows: the set of words corresponding in this sense to solutions of a system of exponential-Diophantine equations is a regular language (i.e. recognizable by a finite automaton). There exists an effective algorithm which calculates this language. This algorithm is constructed in the paper.