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



Regular sequence of power sum symmetric polynomials in polynomial ring.
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. 


