This material has been published in Rend. Sem. Mat. Univ. Padova 121 (2009), 179-199, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by European Mathematical Society Publishing House. This material may not be copied or reposted without explicit permission.

Aldo Conca, Christian Krattenthaler and Junzo Watanabe

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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