Isolated points, duality and residues Original Research Article

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 set membership, variant K[[∂]] in differential operators. We use the correspondence between ideals of R and vector spaces of K[[∂]], stable by derivation and closed for the (∂)-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[∂], 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.