Rendiconti del Seminario Matematico della Università di Padova
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Regular Sequences of Symmetric PolynomialsAldo Conca, Christian Krattenthaler and Junzo Watanabe (1) DIMA - Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146, Genova, Italy
(2) Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, 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.
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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