Isolated points, duality and residues

In this paper, we are interested in the use of duality in effective computations on polynomials. We represent the elements of the dual of the algebra R of polynomials over the field K as formal series is an element of K[[partial derivative]] in differential operators. We use the correspondence between ideals of R and vector spaces of K[[partial derivative]], stable by derivation and closed for the (partial derivative)-adic topology, in order to construct the local inverse system of an isolated point, We propose an algorithm, which computes the orthogonal D of the primary component of this isolated point, by integration of polynomials in the dual space K[partial derivative], with good complexity bounds, Then we apply this algorithm to the computation of local residues, the analysis of real branches of a locally complete intersection curve, the computation of resultants of homogeneous polynomials. (C) 1997 Elsevier Science B.V.