Primary Decomposition of Symmetric Ideals

Abstract: We propose an effective method for primary decomposition of symmetric ideals. Let $K[X]=K[x_1,\ldots,x_n]$ be the $n$-valuables polynomial ring over a field $K$ and $\mathfrak{S}n$ the symmetric group of order $n$. We consider the canonical action of $\mathfrak{S}_n$ on $K[X]$ i.e. $\sigma(f(x_1,\ldots,x_n))=f(x{\sigma(1)},\ldots,x_{\sigma(n)})$ for $\sigma\in \mathfrak{S}_n$. For an ideal $I$ of $K[X]$, $I$ is called {\em symmetric} if $\sigma(I)=I$ for any $\sigma\in \mathfrak{S}_n$. For a minimal primary decomposition $I=Q_1\cap \cdots \cap Q_r$ of a symmetric ideal $I$, $\sigma(I)=\sigma (Q_1)\cap \cdots \cap \sigma(Q_r)$ is a minimal primary decomposition of $I$ for any $\sigma\in \mathfrak{S}_n$. We utilize this property to compute a full primary decomposition of $I$ efficiently from partial primary components. We investigate the effectiveness of our algorithm by implementing it in the computer algebra system Risa/Asir.