Regular sequences of symmetric polynomials
Abstract
Denote by p_k the k-th power sum symmetric polynomial n variables. 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 n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances.
Cite
@article{arxiv.0801.2662,
title = {Regular sequences of symmetric polynomials},
author = {Aldo Conca and Christian Krattenthaler and Junzo Watanabe},
journal= {arXiv preprint arXiv:0801.2662},
year = {2018}
}
Comments
corrected a typo in Lemma 2.4(b)