Computation of graded ideals with given extremal Betti numbers in a polynomial ring

Abstract

Consider a polynomial ring in a finite number of variables over a field of characteristic 0. We implement in CoCoA some algorithms in order to easy compute graded ideals of this ring with given extremal Betti numbers (positions as well as values). More precisely, we develop a package for determining the conditions under which, given two positive integers n,r, 1≤r≤n−1, there exists a graded ideal of a polynomial ring in n variables with r extremal Betti numbers in the given position. An algorithm to check whether an r-tuple of positive integers represents the admissible values of the r extremal Betti numbers is also described. An example in order to show how the package works is also presented.