Primary Decomposition of Symmetric Ideals

We propose an effective method for primary decomposition of symmetric ideals. Let KX=Kx₁, , 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 KX i. e. (f (x₁, , xₙ) ) =f (x (₁), , x (₍) ) for Sₙ. For an ideal I of KX, I is called symmetric if (I) =I for any Sₙ. For a minimal primary decomposition I=Q₁ Qᵣ of a symmetric ideal I, (I) = (Q₁) (Qᵣ) is a minimal primary decomposition of I for any 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.