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.

翻译：我们提出了一种对称理想主分解的有效方法。设$K[X]=K[x_1,\ldots,x_n]$为域$K$上的$n$元多项式环，$\mathfrak{S}_n$为$n$阶对称群。考虑$\mathfrak{S}_n$在$K[X]$上的典范作用，即对于$\sigma\in \mathfrak{S}_n$，有$\sigma(f(x_1,\ldots,x_n))=f(x_{\sigma(1)},\ldots,x_{\sigma(n)})$。若$K[X]$的理想$I$满足对任意$\sigma\in \mathfrak{S}_n$均有$\sigma(I)=I$，则称$I$为对称理想。对于对称理想$I$的一个极小主分解$I=Q_1\cap \cdots \cap Q_r$，任意$\sigma\in \mathfrak{S}_n$作用后得到的$\sigma(I)=\sigma (Q_1)\cap \cdots \cap \sigma(Q_r)$仍是$I$的一个极小主分解。我们利用这一性质，通过部分主分量高效计算$I$的完整主分解。通过在计算机代数系统Risa/Asir中实现该算法，验证了其有效性。