On inverse systems and squarefree decomposition of zero-dimensional polynomial ideals

Authors:
Werner Heiβ;Ulrich Oberst;Franz Pauer
Affiliations:
Institut für Mathematik, Universität Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria;Institut für Mathematik, Universität Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria;Institut für Mathematik, Universität Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria
Venue:
Journal of Symbolic Computation
Year:
2006

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Abstract

Let I be a zero-dimensional ideal in a polynomial ring F[s]:=F[s"1,...,s"n] over an arbitrary field F. We show how to compute an F-basis of the inverse system I^@? of I. We describe the F[s]-module I^@? by generators and relations and characterise the minimal length of a system of F[s]-generators of I^@?. If the primary decomposition of I is known, such a system can be computed. Finally we generalise the well-known notion of squarefree decomposition of a univariate polynomial to the case of zero-dimensional ideals in F[s] and present an algorithm to compute this decomposition.