Related papers: Polynomial recurrence with large intersection over…
We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite…
We study the compositional inverses of some general classes of permutation polynomials over finite fields. We show that we can write these inverses in terms of the inverses of two other polynomials bijecting subspaces of the finite field,…
In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an $IP_r^\star$-strengthened version of the polynomial van der Waerden theorem,…
Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, etc. In this paper, we construct several new classes of permutation…
In this paper, we consider several special polynomials related to associated sequences of polynomials. Finally, we give some new and interesting identities of those polynomials arising from transfer formula for the associated sequences.
Let $\mathbb F_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over $\mathbb F_{q^2}$ with many rational points. The curves…
We offer a new proof that a certain q-analogue of multinomial coeffi- cients furnishes a q-counting of the set of permutations of an associated multiset of positive integers, according to the number of inversions in such arrangements. Our…
We study a general method of the field intersection problem of generic polynomials over an arbitrary field $k$ via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using…
In this paper we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.
We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.
We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is…
By studying $\mathbb{A}^1$-curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete…
In this work we give a full characterization of sets of multiple polynomial recurrence in Weyl systems, which are ergodic unipotent affine transformations on products of tori and finite abelian groups. In particular, we show that measurable…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
Combinatorial interpretation of the fibonomial coefficients recently proposed by the present author results here in combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided with…
A symbolic method for solving linear recurrences of combinatorial and statistical interest is introduced. This method essentially relies on a representation of polynomial sequences as moments of a symbol that looks as the framework of a…
We present three proofs of an observation of Ahmadi on the number of irreducible polynomials over $\text{GF}(2)$ with certain traces and cotraces, the most interesting of which uses an explicit natural bijection. We also present two proofs…
Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite…
We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.