FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2000, VOLUME 6, NUMBER 3, PAGES 649668
Exponential Diophantine equations in rings of positive characteristic
A. Ya. Belov
A. A. Chilikov
Abstract
View as HTML
View as gif image
View as LaTeX source
In this work we prove the algorithmical solvability of
the exponentialDiophantine equations in rings represented by
matrices over fields of positive characteristic.
Consider the system of exponentialDiophantine equations
$$

s
S
i = 1

P_{ij}(n_{1},¼,n_{t})
b_{ij0}a_{ij1}^{n1}b_{ij1}¼a_{ijt}^{nt}b_{ijt} = 0, 

where $b$_{ijk},a_{ijk}
are constants from matrix ring of characteristic $p$, $n$_{i} are
indeterminates.
For any solution $$á n_{1}, ¼ ,n_{t} ñ of the system we
construct the word (over alphabet which contains $pt$ symbols)
$$`a_{0}¼`a_{q},
where $$`a_{i} is
a $t$tuple
$$á
n_{1}^{(i)}, ¼
,n_{t}^{(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 exponentialDiophantine equations is a regular language
(i. e.
recognizible by a finite automaton).
There is an effective algorithm which calculates this language.
All articles are
published in Russian.
Location: http://mech.math.msu.su/~fpm/eng/k00/k003/k00303h.htm
Last modified: December 8, 2000