Related papers: On algebraic integers which are 2-Salem elements i…
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…
It is a classical result that prime numbers of the form $x^2+ny^2$ can be characterized via class field theory for an infinite set of $n$. In this paper we derive the function field analogue of the classical result. Then we apply an…
We survey three methods for proving that the characteristic polynomial of a finite lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on…
A new polynomial identity is found for Dickson polynomials in characteristic 2. The identity is used to prove that the two polynomials $x^{q+1}+x+1/a$ and $C(x)+a$ have the same splitting field over $F$, where $F$ is a field of…
We consider three types of rings of supersymmetric polynomials: polynomial ones $\Lambda_{m,n}$, partially polynomial $\Lambda_{m,n}^{+y}$ and Laurent supersymmetric rings $\Lambda_{m,n}^{\pm}$. For each type of rings we give their…
In symmetric groups, a two-sided cell is the set of all permutations which are mapped by the Robinson-Schensted correspondence on a pair of tableaux of the same shape. In this article, we show that the set of permutations in a two-sided…
Using a special case of Askey's $q$-beta integral evaluation formula, we determine orthogonality relations for the Al-Salam--Carlitz polynomials of type II with respect to a family of measures supported on a discrete subset of $\mathbb R$.…
Let $q$ be a prime power and, for each positive integer $n\ge 1$, let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements. Motivated by the well known concept of normal elements over finite fields, Huczynska et al (2013) introduced…
The aim of this expository article is twofold. The first is to introduce several polynomials of one variable as well as two variables defined on the positive integers with values as congruent numbers. The second is to present connections…
We present a non-standard proof of the fact that the existence of a local (i.e. restricted to a point) characteristic-zero, semi-parametric lifting for a variety defined by the zero locus of polynomial equations over the integers is…
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the…
An interesting discovery in the last two years in the field of mathematical physics has been the exceptional $X_\ell$ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms,…
T. M. Bisgaard proved that the $*$-algebra ${\bf C}[z,\overline{z},1/z\overline{z}]$ has the moment property, that is, each positive linear functional on this $*$-algebra is a moment functional. We generalize this result to polynomials in…
This article studies separating invariants for the ring of multisymmetric polynomials in $m$ sets of $n$ variables over an arbitrary field $\mathbb{K}$. We prove that in order to obtain separating sets it is enough to consider polynomials…
In this paper, we study the dependence of the weak Lefschetz property of algebras defined by a special class of monomials ideals in a polynomial ring with coefficient in a field, to the characteristic of the base field.
We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain $D$. This theorem is relative to any element of $D$. The ingredients are: the arithmetic of Laurent polynomials over $D$, a recursive…
Polynomial time primality tests for specific classes of numbers of the form $k\cdot 2^m \pm 1$ are introduced.
Mayr and Meyer found ideals $J(n,d)$ (in a polynomial ring in $10n+2$ variables over a field $k$ and generators of degree at most $d+2$) with ideal membership property which is doubly exponential in $n$. This paper is a first step in…
It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…
By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $\alpha$ of degree d such that $\alpha^n-1$ is a unit. A similar result is…