Regular Sequences of Symmetric Polynomials

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 $x_1^k+x_2^k+...+x_n^k$. 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 $A\subset {\mathbb{N}}^{\ast }$ of cardinality $n$ such that the set of polynomials $p_a(n)$ with $a\in A$ 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 $abc\equiv 0(\mathrm{mod}6)$. We provide evidence for the conjecture by proving it in several special instances.