Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

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.


share|cite|improve this question

1 Answer 1

up vote 1 down vote accepted

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.

share|cite|improve this answer
Yes, you are correct Gjergji Zaimi. It does follow from lemma 2.2 in Conca, Krattenthaller and Watanabe. Thanks. –  Neeraj May 27 '12 at 6:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.