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.

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

AMATA, Luca;Crupi, Marilena
2019-01-01

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 $n, r$, $1le r le n-1$, 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.
2019
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/3134277
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