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

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 $n, r$, $1le r le n-1$, 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.