Related papers: Polynomial recurrence with large intersection over…
We give a counting formula in terms of modified Hall-Littlewood polynomials and the chromatic quasisymmetric function for the number of points on an arbitrary Hessenberg variety over a finite field. As a consequence, we express the…
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…
We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.
We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
In this paper, we show that the solution to a large class of "tiling" problems is given by a polynomial sequence of binomial type. More specifically, we show that the number of ways to place a fixed set of polyominos on an $n\times n$…
This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partial derivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in a systematic…
This is a short note that explains a problem on polynomial maps over finite fields for non-experts. The problem is: Do there exist odd polynomial automorphisms over the finite fields with 4,8,16,32,64,... elements? The explanation is very,…
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…
We identify a recursive structure among factorizations of polynomial values into two integer factors. Polynomials for which this recursive structure characterizes all non-trivial representations of integer factorizations of the polynomial…
New approach to systems of polynomial recursions is developed based on the Carleman linearization procedure. The article is divided into two main sections: firstly, we focus on the case of uni-variable depth-one polynomial recurrences.…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
We continue the study of intersection bodies of polytopes, focusing on the behavior of $IP$ under translations of $P$. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of $I(P+t)$ can be…
Let $k$ be a field of characteristic $\neq 2$. We survey a general method of the field intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic…
We describe an algorithm that takes as input a complex sequence $(u_n)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound $(v_n)$ such that $|u_n| \leq…
In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…
We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational…
In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…
In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In…