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

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.