ISOLATED POINTS, DUALITY AND RESIDUES

In this paper, we are interested in the use of duality in effective co mputations on polynomials. We represent the elements of the dual of th e algebra R of polynomials over the field K as formal series is an ele ment of K[[partial derivative]] in differential operators. We use the correspondence between ideals of R and vector spaces of K[[partial der ivative]], 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 complex ity bounds, Then we apply this algorithm to the computation of local r esidues, the analysis of real branches of a locally complete intersect ion curve, the computation of resultants of homogeneous polynomials. ( C) 1997 Elsevier Science B.V.