TY - JOUR

T1 - Exponential-Diophantine Equations in rings of positive characteristic

AU - Belov, A.

AU - Chilikov, A.A.

PY - 2000

Y1 - 2000

N2 - In this work we prove the algorithmical solvability of the exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations
s
S
i = 1
Pij(n1,¼,nt) bij0aij1n1bij1¼aijtntbijt = 0,
where bijk,aijk are constants from matrix ring of characteristic p, ni are indeterminates. For any solution á n1, ¼ ,nt ñ of the system we construct the word (over alphabet which contains pt symbols) `a0¼`aq, where `ai is a t-tuple á n1(i), ¼ ,nt(i) ñ, n(i) is the i-th digit in the p-adic representation of n. The main result of this work is: the set of words, corresponding in this sense to the solutions of the system of exponential-Diophantine equations is a regular language (i. e. recognizible by a finite automaton). There is an effective algorithm which calculates this language.

AB - In this work we prove the algorithmical solvability of the exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations
s
S
i = 1
Pij(n1,¼,nt) bij0aij1n1bij1¼aijtntbijt = 0,
where bijk,aijk are constants from matrix ring of characteristic p, ni are indeterminates. For any solution á n1, ¼ ,nt ñ of the system we construct the word (over alphabet which contains pt symbols) `a0¼`aq, where `ai is a t-tuple á n1(i), ¼ ,nt(i) ñ, n(i) is the i-th digit in the p-adic representation of n. The main result of this work is: the set of words, corresponding in this sense to the solutions of the system of exponential-Diophantine equations is a regular language (i. e. recognizible by a finite automaton). There is an effective algorithm which calculates this language.

UR - http://www.emis.ams.org/journals/FPM/eng/k00/k003/k00303h.htm

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VL - 6

SP - 47

EP - 66

JO - Fundamentalnaya i Prikladnaya Matematika (Moscow)

JF - Fundamentalnaya i Prikladnaya Matematika (Moscow)

IS - 3

ER -