Zero-generic initial ideals

Abstract

Given a homogeneous ideal I of a polynomial ring \({A={\mathbb{K}}[X_1,\ldots,X_n]}\) and a monomial order \({\tau}\), we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to \({\tau}\) and denote it with \({{\rm gin}_0(I)}\). When char \({\mathbb{K}=0}\), a zero-generic initial ideal is the usual generic initial ideal. We show that \({{\rm gin}_0(I)}\) is endowed with many interesting properties and, quite surprisingly, it also satisfies Green’s Crystallization Principle, which is known to fail in positive characteristic. Thus, zero-generic initial ideals can be used as formal analogues of generic initial ideals computed in characteristic 0.