Primary Decomposition of Symmetric Ideals
Yuki Ishihara
TL;DR
The paper addresses efficient primary decomposition of symmetric ideals in $K[X]$ by exploiting the action of the symmetric group $\mathfrak{S}_n$. It introduces the quotient set of primary components $\mathcal{Q}[I]$ and an orbit-based decomposition to reduce computation, and develops a symmetric adaptation of the Shimoyama-Yokoyama framework using symmetric separators and saturated separating ideals. The proposed Symmetric Shimoyama-Yokoyama algorithm (symSY) computes minimal primary decompositions from a small set of orbit representatives, with termination guarantees. Experimental results in Risa/Asir demonstrate significant speedups when symmetry is high and the number of components is large, underscoring practical impact and suggesting avenues for extending to other group actions such as $GL(n,K)$ and applications in statistics.
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. $σ(f(x_1,\ldots,x_n))=f(x_{σ(1)},\ldots,x_{σ(n)})$ for $σ\in \mathfrak{S}_n$. For an ideal $I$ of $K[X]$, $I$ is called {\em symmetric} if $σ(I)=I$ for any $σ\in \mathfrak{S}_n$. For a minimal primary decomposition $I=Q_1\cap \cdots \cap Q_r$ of a symmetric ideal $I$, $σ(I)=σ(Q_1)\cap \cdots \cap σ(Q_r)$ is a minimal primary decomposition of $I$ for any $σ\in \mathfrak{S}_n$. 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.