Question

Let $S=\mathbb{C}[x_1,\dots,x_n]$ be a polynomial ring and $p_a=x_1^a+\cdots+x_n^a$ be a power sum symmetric polynomial in $S$. Let $n \geq 3$.

Question: To show $p_m,p_{2m}, \dots,p_{nm}$ forms a regular sequence in $\mathbb{C}[x_1,\dots,x_n]$ for any $m \in \mathbb{N}$. Or equivalently let $I=\langle p_m,p_{2m},\dots,p_{nm} \rangle$ denotes the ideal generated by $p_m,p_{2m},\dots,p_{nm}$. Let $R=S/I$. To show $R$ is a complete intersection.

Facts: It is shown by Conca, Krattenthaller and Watanabe that $p_m,p_{m+1},p_{m+2},p_{m+n-1}$ always forms a regular sequence in $S=\mathbb{C}[x_1,\dots,x_n]$. see http://arxiv.org/abs/0801.2662.

My computer calculation suggests that $p_m,p_{2m},\dots,p_{nm}$ always forms a regular sequence in $\mathbb{C}[x_1,\dots,x_n]$. One may use Newtons identity of power sum to try, although I am unable to conclude.

Thanks
Neeraj

Answer 1

I believe this is true for a very simple reason. See lemma 2.2 in Conca, Krattenthaller and Watanabe.

A non-trivial common zero point of $p_{m},p_{2m},\dots, p_{nm} $ in $n$ variables exists iff there is a non-trivial common zero of $p_{1},p_2,\dots,p_{n}$, but this is absurd. (Every symmetric polynomial would vanish at such a point!)

In general a sequence of power-sums $p_{k_1},\dots,p_{k_n}$ with $k _i=mk' _i$ is regular iff the sequence $p _{k' _1},\dots,p _{k' _n}$ is regular.