Gröbner basis structure of finite sets of points

Abstract

We study the relationship between certain Gröbner bases for zero-dimensional radical ideals, and the varieties defined by the ideals. Such a variety is a finite set of points in an affine n-dimensional space. We are interested in monomial orders that “eliminate ” one variable, say z. Eliminating z corresponds to projecting points in n-space to (n − 1)-space by discarding the z-coordinate. We show that knowing a minimal Gröbner basis under an elimination order immediately reveals some of the geometric structure of the corresponding variety, and knowing the variety makes available information concerning the basis. These relationships can be used to decompose polynomial systems into smaller systems