Isolated points, duality and residues

Bernard Mourrain (1)

1 GALAAD - Geometry, algebra, algorithms

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

Mots clés: polynomial; duality; multiple point; inverse system; effective computation

Domaines: Calcul formel [cs.SC]; Algèbre commutative [math.AC]

Bernard Mourrain. Isolated points, duality and residues. Journal of Pure and Applied Algebra, 1996, 117&118, pp.469--493. ⟨inria-00125278⟩