Related papers: Singular polynomials for the symmetric groups

Given a real univariate degree $d$ polynomial $P$, the numbers $pos_k$ and $neg_k$ of positive and negative roots of $P^{(k)}$, $k=0$, $\ldots$, $d-1$, must be admissible, i.e. they must satisfy certain inequalities resulting from Rolle's…

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and \[\{1,\theta,\theta^2,\ldots, \theta^{N-1}\}\] is a basis for the ring of integers of ${\mathbb Q}(\theta)$,…

In this paper, we study weight representations over the Schr{\"o}dinger Lie algebra $\mathfrak{s}_n$ for any positive integer $n$. It turns out that the algebra $\mathfrak{s}_n$ can be realized by polynomial differential operators. Using…

In the paper, three-dimensional Nijenhuis operators are studied that have differential singularities, i.e., such points at which the coefficients of the characteristic polynomials are dependent. The case is studied in which the…

Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial $K$-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We…

Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for…

We discuss the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an $m\times n$ matrix function $\Phi$ on the unit circle $\mathbb{T}$, when is there a matrix…

This paper studies so-called "null polynomials modulo m", i.e., polynomials with integer coefficients that satisfy f(x)=0 (mod m) for any integer x. The study on null polynomials is helpful to reduce congruences of higher degrees modulo m…

In this paper we study modified kernel polynomials: $u_n(x) = \sum_{k=0}^n c_k g_k(x)$, depending on parameters $c_k>0$, where $\{ g_k \}_0^\infty$ are orthonormal polynomials on the real line. Besides kernel polynomials with $c_k =…

We analyze certain compositions of rational inner functions in the unit polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition is met,…

We give a complete classification of reductive symmetric pairs (g, h) with the following property: there exists at least one infinite-dimensional irreducible (g,K)-module X that is discretely decomposable as an (h,H \cap K)-module. We…

In this paper we introduce and investigate a one-parameter family of polynomials. They are semisymmetric, i.e. symmetric in the variables with odd and even index separately. In fact, the family forms a basis of the space of semisymmetric…

The irreducible characters of the symmetric group are a symmetric polynomial in the eigenvalues of a permutation matrix. They can therefore be realized as a symmetric function that can be evaluated at a set of variables and form a basis of…

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators…

Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint…

Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We…

We consider the set S(n,0) of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ we let $|p|_{0}$ denote the distance from the origin to the zero set of $p'$. We…

Let ${\mathcal B}=\{b_i \}_{i=1}^\infty$ be a fixed sequence of pairwise distinct elements of a number field $k$. Given the integers $2\leq s \leq r$, assuming a quantitative version of Vojta's conjecture on the bounded degree algebraic…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u \in S$ belonging to the socle $\Soc(S/I^{k})$ of…

We prove that if $\sigma \in S_m$ is a pattern of $w \in S_n$, then we can express the Schubert polynomial $\mathfrak{S}_w$ as a monomial times $\mathfrak{S}_\sigma$ (in reindexed variables) plus a polynomial with nonnegative coefficients.…