Exponential diophantine equations in rings of positive characteristic

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.