Primary Decomposition of Symmetric Ideals

Abstract

We propose an effective method for primary decomposition of symmetric ideals. Let $K[X]=K[x_1,\dots ,x_n]$ 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. $\ddot{I}ƒ(f(x_1,\dots ,x_n))=f(x_{\ddot{I}ƒ(1)},\dots ,x_{\ddot{I}ƒ(n)})$ for $\ddot{I}ƒ\in {\mathfrak{S}}_n$. For an ideal $I$ of $K[X]$, $I$ is called {\em symmetric} if $\ddot{I}ƒ(I)=I$ for any $\ddot{I}ƒ\in {\mathfrak{S}}_n$. For a minimal primary decomposition $I=Q_1\cap \cdots \cap Q_r$ of a symmetric ideal $I$, $\ddot{I}ƒ(I)=\ddot{I}ƒ(Q_1)\cap \cdots \cap \ddot{I}ƒ(Q_r)$ is a minimal primary decomposition of $I$ for any $\ddot{I}ƒ\in {\mathfrak{S}}_n$. 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.