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 ⊂ 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 (mod 6). We provide evidence for the conjecture by proving it in several special instances.