Related papers: A systematic approach to Diophantine equations: tw…
This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible…
We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the…
In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a…
An asymptotic formula for the number of prime solutions of a general diagonal system of Diophantine equations is established, contingent on the existence of an appropriate mean value bound and on local solvability. In conjunction with the…
An asymptotic formula which holds almost everywhere is obtained for the number of solutions to the Diophantine inequalities |qA-p|<\psi(|q|), where A is an n by m matrix (m>1) over the field of formal Laurent series with coefficients from a…
We propose a conjecture, similar to Skolem's conjecture, on a Hasse-type principle for exponential diophantine equations. We prove that in a sense the principle is valid for "almost all" equations. Based upon this we propose a general…
In this paper we examine the diophantine equation $x^k-y^k=x-y$ where $k$ is a positive integer $\geq 2$, and consider its applications. While the complete solution of the equation $x^k-y^k=x-y$ in positive rational numbers is already known…
In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the equation $n!=x^2-1$. It is conjectured that this equation has only three solutions, but up to now this is an open problem. Overholt observed…
Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for…
Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…
In the recent years, several polynomial algorithms of a dynamical nature have been proposed to address the graph isomorphism problem. In this paper we propose a generalization of an approach exposed in cond-mat/0209112 and find that this…
We propose a simple method that allows, in one dimension, to solve exactly a wide class of classical stochastic many-body systems far from equilibrium. For the sake of illustration and without loss of generality, we focus on a model that…
We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The…
We give bounds on the number of solutions to the Diophantine equation (X+1/x)(Y+1/y) = n as n tends to infinity. These bounds are related to the number of solutions to congruences of the form ax+by = 1 modulo xy.
We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…
Motivated by Fomin and Zelevinsky's theory of cluster algebras we introduce a variant of the Markov equation; we show that all natural solutions of the equation arise from an initial solution by cluster transformations.
In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by…
In this work we deal with a symbolic approach to the general quadratic polynomial decomposition. By means of a symbolic implementation, we investigate some properties of the components sequences like orthogonality and symmetry. We present…
We obtain asymptotic upper bounds for the number of natural solutions of the following diagonal Diophantine equations in a hypercube with side - $N$ in the paper: $x_1 = x_2^k+...+x_s^k$, $x_1^k = x_2^k+...+x_s^k$, $x_1 = \sum_{j=2}^s…
The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex…