# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 121, 2009, pp. 179–199**

**DOI: 10.4171/RSMUP/121-11**

Regular Sequences of Symmetric Polynomials

Aldo Conca^{[1]}, Christian Krattenthaler

^{[2]}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 *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.

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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