Related papers: Counting solvable $S$-unit equations
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.
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 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 show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a…
We show that there exist arbitrarily large sets $S$ of $s$ prime numbers such that the equation $a+b=c$ has more than $\exp(s^{2-\sqrt{2}-\epsilon})$ solutions in coprime integers $a$, $b$, $c$ all of whose prime factors lie in the set $S$.…
We show that there are arbitrarily large sets $S$ of $s$ primes for which the number of solutions to $a+1=c$ where all prime factors of $ac$ lie in $S$ has $\gg \exp( s^{1/4}/\log s)$ solutions.
Based on combinatorics, we evaluate the upper bounds for the number of solutions to spatially coupled Sudokus, which are popular logic puzzles.
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…
We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.
We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics,…
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(\alpha, \beta)$, we bound the…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
We show that the number of integer solutions for a pair of bilinear equations in at least 2*6 variables has (up to logarithms) the expected upper bound unless there is a structural reason why it is not the case.
Lower bounds for some explicit decision problems over the complex numbers are given.
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.
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…
We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity…
Using the auxiliary field method, a generic upper bound is obtained for the spinless Salpeter equation with two different masses. Analytical results are presented for the cases of the Coulomb and linear potentials when a mass is vanishing.
We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.