相关论文: Symmetric polynomials vanishing on the diagonals s…

We examine the non-symmetric Macdonald polynomials $E_\lambda(x;q,t)$ at $q=1$, as well as the more general permuted-basement Macdonald polynomials. When $q=1$, we show that $E_\lambda(x;1,t)$ is symmetric and independent of $t$ whenever…

We study the following natural question on random sets of points in $\mathbb{F}_2^m$: Given a random set of $k$ points $Z=\{z_1, z_2, \dots, z_k\} \subseteq \mathbb{F}_2^m$, what is the dimension of the space of degree at most $r$…

We study the vanishing sets of slice regular polynomials in several quaternionic variables. We obtain a geometric description of the vanishing sets in two variables, which leads to a new version of the Strong Hilbert Nullstellensatz in the…

Zeros of many ensembles of polynomials with random coefficients are asymptotically equidistributed near the unit circumference. We give quantitative estimates for such equidistribution in terms of the expected discrepancy and expected…

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that…

In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 they have an interpretation in terms of Capelli…

Let $K$ be a field and let $\mathbb N = \{1,2, \dots \}$. Let $R_n=K[x_{ij} \mid 1\le i\le n, j\in \mathbb N]$ be the ring of polynomials in $x_{ij}$ $(1 \le i \le n, j \in \mathbb N)$ over $K$. Let $S_n = Sym (\{1,2, \ldots, n \})$ and…

A singular polynomial is one which is annihilated by all Dunkl operators for a certain parameter value. These polynomials were first studied by Dunkl, de Jeu and Opdam, (Trans. Amer. Math. Soc. 346 (1994), 237-256). This paper constructs a…

Let $\Psi_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $\Psi_n(x)=(x^n-1)/\Phi_n(x)$, with $\Phi_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $\Psi_n(x)$…

Let $N>1$ and let $\Phi_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the…

Like all other knot polynomials, the superpolynomials should be defined in arbitrary representation R of the gauge group in (refined) Chern-Simons theory. However, not a single example is yet known of a superpolynomial beyond symmetric or…

Generalization of the Euler polynomials ${{A}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{m}^{n}}{{x}^{m}}}$ are the polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x…

We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters $(q,t)$ and polynomial in a further two parameters $(u,v)$. We evaluate these polynomials explicitly as a matrix…

Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric…

Let $p$ be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$,…

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the…

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several…

We begin a generalized study of sum-product type phenomenon in different fields by considering pairs $P(x,y)$ and $Q(x,y)$ of two variable polynomials that simultaneously exhibit small symmetric expansion. Our first result is that such…

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready…

Let S=K[x_1,...,x_n] be a polynomial ring. Denote by $p_a$ the power sum symmetric polynomial x_1^a+...+x_n^a. We consider the following two questions: Describe the subsets $A \subset \mathbb{N}$ such that the set of polynomials $p_a$ with…