TY - JOUR

T1 - Exponential diophantine equations in rings of positive characteristic

AU - Chilikov, A. A.

AU - Belov-Kanel, Alexey

N1 - Publisher Copyright:
© 2020 World Scientific Publishing Company.

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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.

AB - 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.

KW - Finite automata

KW - regular languages

UR - http://www.scopus.com/inward/record.url?scp=85082607462&partnerID=8YFLogxK

U2 - 10.1142/S0218216520400027

DO - 10.1142/S0218216520400027

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AN - SCOPUS:85082607462

SN - 0218-2165

VL - 29

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

IS - 2

M1 - 2040001

ER -