Related papers: $S$-unit equation in two variables and Pad\'{e} ap…
We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.
In this paper we improve upon in terms of S the best known effective upper bounds for the solutions of S-unit equations and decomposable form equations.
This paper has two purposes. First we present a new definition of the multivariate Pad\'e approximation, a new fast numerical method. Then numerical solution of the one-dimensional (1D) time-dependent nonlinear Sine-Gordon equation (SGE) is…
We give improved lower bounds for the number of solutions of some $S$-unit equations over the integers, by counting the solutions of some associated linear equations as the coefficients in those equations vary over sparse sets. This method…
We construct explicitly Pad\'e approximations of the second kind for a special class of G-functions. These are then applied to prove a Baker-type lower bound for linear forms in the p-adic values of these functions. Moreover, we consider…
Methods of Pad\'e approximation are used to analyse a multivariate Markov transform which has been recently introduced by the authors, and which is generalizing the well-known in Spectral theory Stieltjes transform (Markov function) of…
We will use Thue-Siegel method, based on Pad\'e approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation $|F(x, y)| = 1$ as well as the inequalities $|F(x, y)| \leq h$, for a…
To generalize the concept of Pad\'e approximation for functions to more than one variable, several definitions have been introduced. All definitions have advantages and disadvantages. The advantages of these approaches has been discussed in…
In this article, we construct new Pad\'{e} approximations for the \emph{product} of binomial functions and powers of logarithmic functions. While several explicit Pad\'{e} approximants are known for powers of exponential functions, binomial…
In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We…
In this paper, we estimate the simultaneous approximation exponents of the values of certain Mahler functions. For this we construct Hermite-Pad\'{e} approximations of the functions under consideration, then apply the functional equations…
We study a variant of the Erd\H os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of angles…
Pad\'e approximants are rational functions whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different…
Exact rational solutions of the generalized Hunter-Saxton equation are obtained using Pad\'e approximant approach for the traveling-wave and self-similarity reduction. A larger class of algebraic solutions are also obtained by extending a…
In this paper, we prove two results related to the solutions of norm form equations. Firstly, we give a finiteness result for sums of terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations.…
We consider the problem of finding approximate analytical solutions for nonlinear equations typical of physics applications. The emphasis is on the modification of the method of Pad\'e approximants that are known to provide the best…
We show how to effectively solve 5-term $S$-unit equations when the set of primes $S$ has cardinality at most 3, and use this to provide an explicit answer to an old question of D.J. Newman on representations of integers as sums of…
Let $P$ be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions $n\leq N$ to $n! = P(x)$ which yields a power saving over the trivial bound. In particular, this applies…
Pad'e approximants are used to improve the convergence behavior of perturbative results in massless scalar and gauge field theories at finite temperature.
In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…