Rendiconti del Seminario Matematico della Università di Padova
Full-Text PDF (154 KB) | Metadata | Table of Contents | RSMUP summary
Published online: 2009-06-30
Regular Sequences of Symmetric PolynomialsAldo Conca, Christian Krattenthaler and Junzo Watanabe (1) Università di Genova, Italy
(2) Universität Wien, Austria
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 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 pa(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 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.
No keywords available for this article.
Conca Aldo, Krattenthaler Christian, Watanabe Junzo: Regular Sequences of Symmetric Polynomials. Rend. Sem. Mat. Univ. Padova 121 (2009), 179-199. doi: 10.4171/RSMUP/121-11