Elimination Ideal

An elimination ideal is the intersection of an ideal in a polynomial ring with the polynomial ring in a specified subset of its variables. For an ideal over a field , its th elimination ideal is

where (Cox et al. 1997, p. 113). Its members are precisely the polynomials in that do not involve , ..., .

The elimination theorem gives a computational method. If is a Gröbner basis of for lexicographic order , then is a Gröbner basis of (Cox et al. 1997, p. 113). Thus one retains just the members of involving the remaining variables.

For example, the ideal in gives

This computes the implicit equation of the curve parameterized by and . More generally, implicitization uses elimination ideals to find the polynomial equations satisfied by a parameterization.

The projection of every solution of the original polynomial equations satisfies the elimination equations, but the converse can fail. For in , eliminating gives . The corresponding zero set is all of , whereas the projected solutions have .