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 Pk(n) the power sum symmetric polynomial in n variables x1k+x2k+... +xnk. The interpretation of the g-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 pa(n) with a e A is a regular sequence. We prove that a necessary condition is that n! divides the product 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