Value distribution of elementary symmetric polynomials and its perturbations over finite fields

初等对称多项式与一个可能的单射映射家族在整数划分上的应用

Elementary symmetric polynomials and a potentially injective family of maps on partitions

本文通过无限例证反驳Ballantine及其合作者关于整数划分上pre_k映射单射性的猜想，研究了这些映射间的相互关系，并提供了三种子情况的替代证明。

The elementary symmetric functions of a reciprocal polynomial sequence

Erdös and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $n\geq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functio…

The elementary symmetric functions of reciprocals of the elements of arithmetic progressions

Let $a$ and $b$ be positive integers. In 1946, Erdős and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. In this paper, we show that for any integer $k$ with $1\le k\le n$, the $k$-th elementary symmetric function of $1/b, 1/(a+b),..., 1/(an-a+b)$ is not an integer except that either $b=n=k=1$ and $a\ge 1$, or $a=b=1, n=3$ and $k=2$. This refines the Erdős-Niven theorem and answers an open problem raised by Chen and Tang in 2012.

关于 $\{1,1/2,\dots,1/n\}\backslash\{1/i\}$ 的初等对称函数

On the Elementary Symmetric Functions of $\{1,1/2,\dots,1/n\}\backslash\{1/i\}$

本文研究从调和数列前 $n$ 项中删去 $1/i$ 后所得集合的初等对称函数的整数性，证明当 $n\geqslant5$ 时均非整数，仅 $n=i=2$ 和 $n=i=4$ 为例外。

Extensions of local fields and elementary symmetric polynomials

Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension of degree $n=up^ν$. Let $σ_1,\dots,σ_n$ denote the $K$-embeddings of $L$ into a separable closure $K^{sep}$ of $K$. For $1\le h\le n$ let $e_h(X_1,\dots,X_n)$ denote the $h$th elementary symmetric polynomial in $n$ variables, and for $α\in L$ set $E_h(α) =e_h(σ_1(α),\dots,σ_n(α))$. Set $j=\min\{v_p(h),ν\}$. We show that for $r\in\mathbb{Z}$ we have $E_h(\mathcal{M}_L^r)\subset \mathcal{M}_K^{\lceil(i_j+hr)/n\rceil}$, where $i_j$ is the $j$th index of inseparability…

Extremely symmetric primes

We introduce extremely symmetric primes and provide some elementary properties of these.