# Regular Sequences of Symmetric Polynomials

### Aldo Conca

Università di Genova, Italy### Christian Krattenthaler

Universität Wien, Austria### Junzo Watanabe

Tokai University, Hiratsuka, Japan

## Abstract

A set of $n$ homogeneous polynomials in $n$ variables is a regular sequence if the associated polynomial system has only the obvious solution $(0,0,...,0)$. Denote by $p_{k}(n)$ the power sum symmetric polynomial in $n$ variables $x_{1}+x_{2}+...+x_{n}$. The interpretation of the $q$-analogue of the binomial coefficient as Hilbert function leads us to discover that $n$ consecutive power sums in $n$ variables form a regular sequence. We consider then the following problem: describe the subsets $A⊂N_{∗}$ of cardinality $n$ such that the set of polynomials $p_{a}(n)$ with $a∈A$ is a regular sequence. We prove that a necessary condition is that $n!$ divides the product of the degrees of the elements of $A$. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already for $n=3$. Given positive integers $a<b<c$ with gcd $(a,b,c)=1$, we conjecture that $p_{a}(3),p_{b}(3),p_{c}(3)$ is a regular sequence if and only if $abc≡0(mod6)$. We provide evidence for the conjecture by proving it in several special instances.

## Cite this article

Aldo Conca, Christian Krattenthaler, Junzo Watanabe, Regular Sequences of Symmetric Polynomials. Rend. Sem. Mat. Univ. Padova 121 (2009), pp. 179–199

DOI 10.4171/RSMUP/121-11