対称イデアルの準素分解アルゴリズム

Abstract

This article discusses primary decompositions of symmetric ideals. Let 𝐾[𝑋] = 𝐾[𝑥₁, ..., 𝑥ₙ] be the 𝑛-variables polynomial ring over a filed 𝐾 and 𝕾ₙ the symmetric group of order 𝑛. Here, 𝑆ₙ acts on 𝐾[𝑋] by σ(𝑓(𝑥₁, ..., 𝑥ₙ)) = 𝑓(𝑥[σ]₍₁₎, ..., 𝑥[σ]₍ₙ₎) for 𝑓 ∈ 𝐾[𝑋] and σ ∈ 𝕾ₙ. An ideal 𝐼 of 𝐾[𝑋] is called a symmetric ideal if σ(𝐼) = 𝐼 for any σ ∈ 𝕾 . For a primary decomposition 𝐼 = 𝑄₁ ∩・・・∩ 𝑄ᵣ of a symmetric ideal 𝐼, 𝐼 = σ(𝑄₁)∩・・・∩σ(𝑄ᵣ) is also a primary decomposition of 𝐼. We introduce an efficient algorithm to compute a primary decomposition of a symmetric ideal.