Related papers: The larger sieve and polynomial congruences
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as…
We show that the large sieve is optimal for almost all exponential sums, thus proving a conjecture by Erd\"os and Renyi.
We improve the known upper bound for short exponential sums and increase the range on which a sharp upper bound is known.
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
We provide infinitely many solutions of a Dirichlet problem on balls.
Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are…
In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.
Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equiv\lambda \pmod{p}$, where $p$ is a prime and $x,y$ lie in certain intervals…
A category which generalises to higher dimensions many of the features of the Temperley-Lieb category is introduced.
We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…
We prove upper and lower bounds for all the coefficients in the Hilbert Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution over $R$. The bounds are in terms of the minimal and the maximal shifts in the resolution…
In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a…
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.
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint…
We improve the Cauchy radius of both scalar and matrix polynomials, which is an upper bound on the moduli of the zeros and eigenvalues, respectively, by using appropriate polynomial multipliers.
We provide explicit bounds on the difference of heights of isogenous Drinfeld modules. We derive a finiteness result in isogeny classes. In the rank 2 case, we also obtain an explicit upper bound on the size of the coefficients of modular…
By employing certain extended classical summation theorems, several surprising \pi and other formulae are displayed.