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arXiv · 1712.00508

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

A. A. Chilikov·A. Ya. Belov
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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) ${\overline α_0},\dots,{\overline α_q}$ where ${\overline α_i}$ 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.

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BibTeXRIS

A. A. Chilikov, A. Ya. Belov. 2019-06-05. Normal basises of algebras and Exponential Diophantine equations in rings of positive characteristic. https://doi.org/10.1142/s0218216520400027

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