|
Fundamentalnaya i Prikladnaya Matematika, 2000, Volume 6, Issue 3, Pages 649–668
(Mi fpm496)
|
|
|
|
This article is cited in 1 scientific paper (total in 1 paper)
Exponential Diophantine equations in rings of positive characteristic
A. Ya. Belova, A. A. Chilikovb a House of scientific and technical work of youth
b M. V. Lomonosov Moscow State University
Abstract:
In this work we prove the algorithmical solvability of the exponential-Diophan-tine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations
$$
\sum_{i=1}^{s}P_{ij}(n_1,\ldots,n_t)b_{ij0}a_{ij1}^{n_1}b_{ij1}\ldots a_{ijt}^{n_t}b_{ijt}=0
$$
where $b_{ijk},a_{ijk}$ are constants from matrix ring of characteristic $p$, $n_i$ are indeterminates. For any solution $\langle n_1,\ldots,n_t \rangle$ of the system we construct the word (over alphabet which contains $p^t$ symbols) $\overline\alpha_0\ldots\overline\alpha_q$, where $\overline\alpha_i$ is a $t$-tuple $\langle n_1^{(i)},\ldots,n_t^{(i)}\rangle$, $n^{(i)}$ 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.
Received: 01.03.1998
Citation:
A. Ya. Belov, A. A. Chilikov, “Exponential Diophantine equations in rings of positive characteristic”, Fundam. Prikl. Mat., 6:3 (2000), 649–668
Linking options:
https://www.mathnet.ru/eng/fpm496 https://www.mathnet.ru/eng/fpm/v6/i3/p649
|
Statistics & downloads: |
Abstract page: | 570 | Full-text PDF : | 183 | First page: | 1 |
|