Related papers: Greatest common divisors for polynomials in almost…

We generalize the solution of linear recurrence relations from fields to central division algebras, adapting the standard tools of companion matrices and characteristic polynomials to the non-commutative setting. We then solve linear…

In their recent article, Min Ru and Paul Vojta, among other things, proved the so-called general theorem (arithmetic part) which can be viewed as an extension of Schmidt's subspace theorem. In this note, we extend their result by replacing…

We prove upper and lower bounds for leading coefficient of Kolchin dimension polynomial of systems of partial linear differential equations in codimension two.

We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…

We prove several bounds on the number of incidences between two sets of multivariate polynomials of bounded degree over finite fields. From these results, we deduce bounds on incidences between points and multivariate polynomials, extending…

We define new generalized factorials in several variables over an arbitrary subset $\underline{S} \subseteq R^n,$ where $R$ is a Dedekind domain and $n$ is a positive integer. We then study the properties of the fixed divisor…

As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel…

We give new combinatorial proofs of known almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask, whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We…

The principal aim of this paper is to extend Abel's theorem to the setting of complex supermanifolds of dimension 1|q over a finite-dimensional local supercommutative C-algebra. The theorem is proved by establishing a compatibility of Serre…

Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials,…

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with…

We show asymptotic upper and lower bounds for the greatest common divisor of N and $\sigma(N)$. We also show that there are infinitely many integers N with fairly large g.c.d. of N and $\sigma(N)$.

Let $\mathcal{A}$ and $\mathcal{B}$ be sets of polynomials of degree $n$ over a finite field. We show, that if $\mathcal{A}$ and $\mathcal{B}$ are large enough, then $A+B$ has an irreducible divisor of large degree for some…

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive…

We deduce a special case of a theorem of M. Haiman concerning alternating polynomials in 2n variables from our results about almost commuting variety, obtained earlier in a joint work with W.-L. Gan.

In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric…

A new polynomial sieve is presented and used to show that almost all integers have at most one representation as a sum of two values of a given polynomial of degree at least 3.

By studying a categorification of the antisymmetriser quasi-idempotent in the Hecke algebra, we derive a closed formula for the Jones-Wenzl idempotent in the Temperley-Lieb algebra. In particular, we show that when the idempotent is…

We examine the maximal number of zeros a polynomial of degree at most n with constrained coefficients may have at 1. Our results are essentially sharp and extend earlier results of this variety. An interesting connection to certain…

For a wide class of sequences of integer domains $\mathcal{D}_n\subset\mathbb{N}^d$, $n\in\mathbb{N}$, we prove distributional limit theorems for $F(X_1^{(n)},\ldots,X_d^{(n)})$, where $F$ is a multivariate multiplicative function and…