Exponential diophantine equations in rings of positive characteristic

Chilikov, A.A.; Belov-Kanel, A.

Авторы: Chilikov A.A., Alexey Belov-Kanel
Журнал: Journal of Knot Theory and its Ramifications
Том: 29
Номер: 2
Год издания: 2020
Издательство: World Scientific Publishing Co
Местоположение издательства: Singapore
DOI: 10.1142/S0218216520400027
Аннотация: 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.
Добавил в систему: Белов Алексей Яковлевич

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