# Regular sequences of symmetric polynomials

### (17 pages)

Abstract. A set of n homogeneous polynomials in n variables is called 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 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 of N* of cardinality n such that the set of polynomials pa(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 a 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 pa(3),pb(3),pc(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.

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