Exponential diophantine equations in rings of positive characteristic

A. A. Chilikov, Alexey Belov-Kanel

Research output: Contribution to journalArticlepeer-review


In this paper, we prove an algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations i=1sP ij(n1,..,nt)bij0aij1n1bij1..aijtntbijt = 0 where bijk,aijk are constants from matrix ring of characteristic p, ni are indeterminates. For any solution (n1,..,nt) of the system we construct a word (over an alphabet containing pt symbols) ᾱ0,..,ᾱq where ᾱi is a t-tuple (n1(i),..,n t(i)), n(i) is the ith digit in the p-adic representation of n. The main result of this paper is following: 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 algorithm which calculates this language. This algorithm is constructed in the paper.

Original languageEnglish
Article number2040001
JournalJournal of Knot Theory and its Ramifications
Issue number2
StatePublished - 1 Feb 2020
Externally publishedYes

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  • Finite automata
  • regular languages


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