Related papers: Symmetric polynomials and non-finitely generated $…

We identify a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, for any collection of $n$ non-negative real numbers $a_1, \dots, a_n \in…

Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear…

Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote…

For a bivariate $P(x,y) \in \mathbb{R}[x,y]\setminus (\mathbb{R}[x] \cup \mathbb{R}[y])$, our first result shows that for all finite $A \subseteq \mathbb{R}$, $|P(A,A)|\geq \alpha|A|^{5/4}$ with $\alpha =\alpha(\mathrm{deg} P) \in…

The set $S(n)$ of all elementary symmetric polynomials in $n$ variables is a minimal generating set for the algebra of symmetric polynomials in $n$ variables, but over a finite field ${\mathbb F}_q$ the set $S(n)$ is not a minimal…

We consider a certain left action by the monoid $SL_2(\mathbf{N}_0)$ on the set of divisor pairs $\mathcal{D}_f := \{ (m, n) \in \mathbf{N}_0 \times \mathbf{N}_0 : m \lvert f(n) \}$ where $f \in \mathbf{Z}[x]$ is a polynomial with integer…

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $J=J(G)$ is its cover ideal. We…

We show that if a polynomial $f\in \mathbb{R}[x_1,\ldots,x_n]$ is nonnegative on a closed basic semialgebraic set $X=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\ldots,g_r (x)\ge 0\}$, where $g_1,\ldots,g_r\in\mathbb{R}[x_1,\ldots,x_n]$, then $f$ can be…

The triangle of sorted binomial coefficients $\left\langle {n \atop k} \right\rangle = \binom{n}{\lfloor \frac{n - k}{2} \rfloor}$ for $0 \leq k \leq n$ has appeared several times in recent combinatorial works but has evaded dedicated…

This paper considers a finite group $G$ acting linearly on the variables $V$ of a polynomial algebra, or an exterior algebra, or superpolynomial algebra with both commuting and anticommuting variables. In this setting, the Hilbert series…

We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group $S_n$ and are consistent as $n$ varies. The extreme infinitely spherically symmetric permutation-valued processes are identified…

Motivated by a question in Schubert calculus, we study the interplay of quasisymmetric polynomials with the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahedra. Divided…

The ring of dual numbers over a ring $R$ is $R[\alpha] = R[x]/(x^2)$, where $\alpha$ denotes $x+(x^2)$. For any finite commutative ring $R$, we characterize null polynomials and permutation polynomials on $R[\alpha]$ in terms of the…

In 2011, Neil Hindman proved that for every natural number $n,m$ the polynomial \begin{equation*} \sum_{i=1}^{n} x_{i}-\prod\limits_{j=1}^{m} y_{j} \end{equation*} has monochromatic solutions for every finite coloration of $\mathbb{N}$. We…

Consider the algebra Q<<x_1,x_2,...>> of formal power series in countably many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...) of symmetric functions in noncommuting variables consists of all elements invariant…

We exploit the $6D, {\cal N}=(1,0)$ and ${\cal N}=(1,1)$ harmonic superspace approaches to construct the full set of the maximally supersymmetric on-shell invariants of the canonical dimension ${\bf d=12}$ in $6D, \,{\cal N}=(1,1)$…

Let $G$ be a subgroup of $S_n$, the symmetric group of degree $n$. For any field $k$, $G$ acts naturally on the rational function field $k(x_1,x_2,\ldots,x_n)$ via $k$-automorphisms defined by $\sigma\cdot x_i=x_{\sigma(i)}$ for any…

We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider…

It is known that the elementary symmetric polynomials $e_k(x)$ have the property that if $ x, y \in [0,\infty)^n$ and $e_k(x) \leq e_k(y)$ for all $k$, then $||x||_p \leq ||y||_p$ for all real $0\leq p \leq 1$, and moreover $||x||_p \geq…

Let $R$ be a finite non-commutative ring with $1\ne 0$. By a polynomial function on $R$, we mean a function $F\colon R\longrightarrow R$ induced by a polynomial $f=\sum\limits_{i=0}^{n}a_ix^i\in R[x]$ via right substitution of the variable…