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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, Krathentheller Krattenthaller and Watababe 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
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Let $S=\mathbb{C}[x_1,\dots,x_4]$ S=\mathbb{C}[x_1,\dots,x_n]$ be a polynomial ring and $p_a=x_1^a+\cdots+x_4^a$ p_a=x_1^a+\cdots+x_n^a$ be a power sum symmetric polynomial in $S$. Let $n \geq 3$.

Question: To show $p_m,p_{2m},p_{3m},p_{4m}$ p_m,p_{2m}, \dots,p_{nm}$ forms a regular sequence in $\mathbb{C}[x_1,\dots,x_4]$ \mathbb{C}[x_1,\dots,x_n]$ for any $m \in \mathbb{N}$. Or equivalently let $I=\langle p_m,p_{2m},p_{3m},p_{4mp_m,p_{2m},\dots,p_{nm} \rangle$ denotes the ideal generated by $p_m,p_{2m},p_{3m}$, and $p_{4m}$. p_m,p_{2m},\dots,p_{nm}$. Let $R=S/I$. To show $R$ is a complete intersection.

Facts: It is shown by Conca, Krathentheller and Watababe that $p_m,p_{m+1},p_{m+2},p_{m+3}$ p_m,p_{m+1},p_{m+2},p_{m+n-1}$ always forms a regular sequence in $S=\mathbb{C}[x_1,\dots,x_4]$. S=\mathbb{C}[x_1,\dots,x_n]$. see http://arxiv.org/abs/0801.2662.

My computer calculation suggests that $p_m,p_{2m},p_{3m},p_{4m}$ p_m,p_{2m},\dots,p_{nm}$ always forms a regular sequence in $\mathbb{C}[x_1,\dots,x_4]$. \mathbb{C}[x_1,\dots,x_n]$. One may use Newtons identity for of power sum to try, although I am unable to conclude.

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

Let $S=\mathbb{C}[x_1,\dots,x_4]$ be a polynomial ring and $p_a=x_1^a+\cdots+x_4^a$ be a power sum symmetric polynomial in $S$.

Question: To show $p_m,p_{2m},p_{3m},p_{4m}$ forms a regular sequence in $\mathbb{C}[x_1,\dots,x_4]$ for any $m \in \mathbb{N}$. Or equivalently let $I=\langle p_m,p_{2m},p_{3m},p_{4m} \rangle$ denotes the ideal generated by $p_m,p_{2m},p_{3m}$, and $p_{4m}$. Let $R=S/I$. To show $R$ is a complete intersection.

Facts: It is shown by Conca, Krathentheller and Watababe that $p_m,p_{m+1},p_{m+2},p_{m+3}$ always forms a regular sequence in $S=\mathbb{C}[x_1,\dots,x_4]$. see http://arxiv.org/abs/0801.2662.

My computer calculation suggests that $p_m,p_{2m},p_{3m},p_{4m}$ always forms a regular sequence in $\mathbb{C}[x_1,\dots,x_4]$. One may use Newtons identity for power sum, although I am unable to conclude.

Thanks
Neeraj