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

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(perpendicular to) of I. We describe the F[s]-module I(perpendicular to) generators and relations and characterise the minimal length of a system of F[s]-generators of I(perpendicular to). If the primary decomposition of I is known, such a system can be computed. Finally we generalise the wellknown 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. (c) 2005 Elsevier Ltd. All rights reserved.