Related papers: A Class of Binomial Permutation Polynomials
A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in…
We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions…
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…
A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number--theoretical properties. A new class of…
In this paper, we give a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of…
In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.
We discuss various aspects of representation of a polynomial as a sum of monomials (for example, uniqueness of such representation and related estimations).
Four recursive constructions of permutation polynomials over $\gf(q^2)$ with those over $\gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over…
We give necessary and sufficient existence criteria, and methods for finding, continuous solutions of linear equations whose coefficients are polynomials.
We introduce colored Jones polynomials of nanowords and their categorification. We also prove the existence of a Khovanov-type bicomplex which has three grades.
For an odd prime power $q$ satisfying $q\equiv 1\pmod 3$ we construct totally $2(q-1) $ permutation polyomials, all giving involutory permutations with exactly $ 1+ \frac{q-1}3$ fixed points. Among them $(q-1)$ polynomials are trinomials,…
We prove a binomial formula for Macdonald polynomials and consider applications of it.
It is known that the polyomino ideal of a simple polyomino is a prime ideal. A new class of nonsimple polyominoes $\Pc$ for which the polyomino ideal $I_{\Pc}$ is a prime ideal will be presented.
In this paper, we consider the matrix polynomial obtained by using bi-periodic Fibonacci matrix polynomial. Then, we give some properties and binomial transforms of the new matrix polynomials.
Motivated by many recent constructions of permutation polynomials over $\mathbb{F}_{q^2}$, we study permutation polynomials over $\mathbb{F}_{q^3}$ in terms of their coefficients. Based on the multivariate method and resultant elimination,…
We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\geq 2$ and $a\in \mathbb{F}_{q^e}^*$. As a consequence, we obtain a complete characterization of such permutation…
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.
Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\Bbb F_{q^2}$. This result allows us to solve a related…
We discuss several constructions of swap polynomials, that is 2--tensor valued matrix polynomials which are multiples of the swap or switch operator.