Initial Ideal

The initial ideal of an ideal in a polynomial ring over a field, with respect to a fixed monomial order , is the monomial ideal generated by the leading monomials of all its nonzero elements,

(1)

The coefficient of a leading term is invertible in the field, so using leading terms gives the same ideal.

A finite subset of the nonzero elements of is a Gröbner basis for iff

(2)

An arbitrary generating set need not have this property. For example, in is the ideal . In lexicographic order with , both displayed generators have leading monomial , whereas .

For a real weight vector , the weighted initial ideal is defined using initial forms,

(3)

It need not be a monomial ideal. For instance, if and , then (Fukuda et al. 2007).