Primary Decomposition of Symmetric Ideals

We propose an effective method for primary decomposition of symmetric ideals. Let K[X]=K[x₁,…,xₙ] be the n-valuables polynomial ring over a field K and Sₙ the symmetric group of order n. We consider the canonical action of Sₙ on $K[X]$ i.e. σ(f(x₁,…,xₙ))=f(xσ(1),…,xσ(n)) for σ∈ Sₙ. For an ideal I of $K[X]$, I is called { symmetric} if σ(I)=I for any σ∈ Sₙ. For a minimal primary decomposition I=Q₁∩ ⋯ ∩ Qᵣ of a symmetric ideal I, σ(I)=σ (Q₁)∩ ⋯ ∩ σ(Qᵣ) is a minimal primary decomposition of I for any σ∈ Sₙ. 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.