# Regular Sequences of Symmetric Polynomials

### Aldo Conca

Università di Genova, Italy### Christian Krattenthaler

Universität Wien, Austria### Junzo Watanabe

Tokai University, Hiratsuka, Japan

## Abstract

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.

## Cite this article

Aldo Conca, Christian Krattenthaler, Junzo Watanabe, Regular Sequences of Symmetric Polynomials. Rend. Sem. Mat. Univ. Padova 121 (2009), pp. 179–199

DOI 10.4171/RSMUP/121-11