Regular sequence of power sum symmetric polynomials in polynomial ring. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:09:04Z http://mathoverflow.net/feeds/question/98075 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98075/regular-sequence-of-power-sum-symmetric-polynomials-in-polynomial-ring Regular sequence of power sum symmetric polynomials in polynomial ring. Neeraj 2012-05-26T23:44:41Z 2012-05-27T00:18:56Z <p>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 $n \geq 3$.</p> <p>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.</p> <p>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 <a href="http://arxiv.org/abs/0801.2662" rel="nofollow">http://arxiv.org/abs/0801.2662</a>. </p> <p>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. </p> <p>Thanks<br> Neeraj</p> http://mathoverflow.net/questions/98075/regular-sequence-of-power-sum-symmetric-polynomials-in-polynomial-ring/98076#98076 Answer by Gjergji Zaimi for Regular sequence of power sum symmetric polynomials in polynomial ring. Gjergji Zaimi 2012-05-27T00:16:14Z 2012-05-27T00:16:14Z <p>I believe this is true for a very simple reason. See lemma 2.2 in Conca, Krattenthaller and Watanabe.</p> <p>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!)</p> <p>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. </p>