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[s1,…,sn] 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.