Related papers: Bounds for the solutions of $S$-unit equations and…
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 article, we use Pad\'{e} approximations constructed for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation. Combining explicit formulae of these Pad\'{e} approximants with a simple…
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…
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 give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+\epsilon})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the…
Let $T$ be a positive integer, and $\mathcal{M}$ be a finitely presented module over the Laurent polynomial ring $\mathbb{Z}_{/T}[X_1^{\pm}, \ldots, X_N^{\pm}]$. We consider S-unit equations over $\mathcal{M}$: these are equations of the…
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…
In this survey paper, we study the optimal regularity of solutions to uniformly degenerate elliptic equations in bounded domains and establish the H\"older continuity of solutions and their derivatives up to the boundary.
We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…
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…
Based on combinatorics, we evaluate the upper bounds for the number of solutions to spatially coupled Sudokus, which are popular logic puzzles.
We show that a particular parameter plays a vital role in the resolution of S-unit equations, at the stage where LLL-reduction is applied. We define the notion of optimal system of fundamental S-units (with respect to this parameter), and…
We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds…
Some new Frobenius norm bounds of the unique solution to certain structured Sylvester equation are derived. Based on the derived norm upper bounds, new multiplicative perturbation bounds are provided both for subunitary polar factors and…
We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees…
We consider the integration of two-dimensional, piecewise constant functions with respect to copulas. By drawing a connection to linear assignment problems, we can give optimal upper and lower bounds for such integrals and construct the…
This paper is concerned with the existence of positive solutions of second-order impulsive differential equations with integral boundary conditions on an infinite interval. As an application, an example is given to demonstrate our main…
In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying…
Motivated by the effective bounds of ordinary differential equations, we prove an effective version of uniform bounding for partial differential fields with commuting derivations. More precisely, we provide an upper bound for the size of…
In this paper, we prove the finiteness of the number of integer solutions of the decomposable form inequalities. We also study the number of integer solutions of a sequence of decomposable form inequalities.