Primary Decomposition of Symmetric Ideals
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910411973984256 |
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| author | Ishihara, Yuki |
| author_facet | Ishihara, Yuki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10482 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primary Decomposition of Symmetric Ideals Ishihara, Yuki Commutative Algebra Symbolic Computation 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. |
| title | Primary Decomposition of Symmetric Ideals |
| topic | Commutative Algebra Symbolic Computation |
| url | https://arxiv.org/abs/2404.10482 |