Finding sparse systems of parameters

Abstract

For several computational procedures such as finding radicals and Noether normalizations, it is important to choose as sparse as possible a system of parameters in a polynomial ideal or modulo a polynomial ideal. We describe new strategies for these tasks, thus providing solutions to problems (1) and (2) posed by D. Eisenbud et al. [Invent. Math. 110 (1992) 207-236]. To accomplish the first task we introduce a notion of “setwise complete intersection”. We prove that a set of monomials generating an ideal of codimension c in a polynomial ring can be partitioned into c disjoint sets forming a setwise complete intersection, although the corresponding result is false for arbitrary sets of polynomials. We reduce the general case to the monomial case by a deformation argument. For homogeneous ideals the output is homogeneous. Our analysis of the second task is based on a concept of Noether complexity for homogeneous ideals and its characterization in terms of Chow forms.