Abstract :
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.
