WHEN ARE SYMMETRIC IDEALS MONOMIAL?

Abstract

We study conditions on a polynomial such that the ideal generated by its orbit under the symmetric group action is a monomial ideal or has a monomial radical. If the polynomial is homogeneous, we expect the ideal to have a monomial radical if the coefficients are sufficiently general with respect to the support of the polynomial. We prove this in the case of a symmetric support set in sufficiently many variables over characteristic zero. If in addition the polynomial has only square-free terms and its coefficients do not sum to zero, then in a larger polynomial ring the ideal itself is square-free monomial. This has implications also for symmetric ideals of the infinite polynomial ring.