Abstract
At how many points is it possible to prescribe values to a polynomial of given degree, together with derivatives up to order, say, t-1? We solve this problem in case of polynomials in two variables for t=2 and t=3 and in case of polynomials in three variables for t=2. Proofs develop in the frame of modern projective geometry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliographie
Arbarello,E.-Cornalba,M.,A few remarks about the variety of irreducible plane curves of given degree and genus, Ann.Scient. Ec.Norm.Sup.16 (1983), 467–488
Ballico,E.,On the postulation of disjoint rational curves in a projective space, Preprint, Pise 1984
-,Homogeneous ideals of points,Preprint,Pise 1984
Ballico,E.-Ellia,Ph.,Generic curves of small genus in ℙ 3 are of maximal rank, Math.Ann.264 (1983) 211–225
-,The maximal rank conjecture for non special curves in ℙ 3, Invent. Math., à paraître
Briançon,J.-Galligo,A.,Déformations distinguées d'un point de ℂ 2 ou ℝ 2, In Astérisque 7/8 (1973), 129–138
Chudnovsky,G.V.,Finite sets of singular points of complex hypersurfaces, CRAS ser.A286 (1978), 1133–1135
—,Singular points on complex hypersurfaces and multidimensional Schwarz lemma, Sem. D.P.P., Prog, in Math.12 (1981), 29–69
Demailly,J.P.,Formules de Jensen en plusieurs variables et applications arithmétiques, Bull. SMF110 (1982), 75–102
Eisenbud,D.-Harris,J.,Divisors on general curves and cuspidal rational curves, Invent.Math74 (1983), 371–418
Ellingsrud,G.,Sur le schéma de Hilbert des variétés de codimension 2 dans 1c e à cône de Cohen-Macaulay, Ann.Sci. Ec.Norm.Sup. Paris8 (1975), 423–432
Esnault,H.-Viehweg,E.,Sur une minoration du degré d'hypersurfaces s'annulant en certains points, Math.Ann.263 (1983), 75–86
Grothendieck,A.,Techniques de construction en géométrie algébrique. Les schémas de Hilbert, Sem. Bourbaki, exposé221, (1960–61)
Gruson,L.-Peskine,Ch.,Genre des courbes de l'espace projectif I. In Algebraic Geometry, Troms∅ 1977, L.N. in Math.687 (1978), 31–59
Hartshorne,R.,Algebraic Geometry, Graduate Texts in Math.52, Springer Verlag, New York (1977) XVI+496pp.
—,On the classification of Space Curves, In Vector Bundles and Differential Equations, Prog. in Math.7 (1980), 83–112
Hartshorne,R.-Hirschowitz,A.,Droites en position générale dans l'espace projectif, In Algebraic Geometry, Proc. La Rabida 1981, L.N. in Math.961 (1982), 169–189
—,Cohomology of a general instanton bundle, Ann.Scient. Ec.Norm.Sup.15 (1982), 365–390
-,Smoothing Algebraic Space Curves, In Week of Algebraic Geometry, Proc. Barcelona 1983, L.N., à paraître
-,Courbes rationnelles et droites en position générale, Preprint Nice 1984
-,Nouvelles courbes de bon genre via la cohomologie des faisceaux réflexifs, en préparation
Hirschowitz,A,Sur la postulation générique des courbes rationnelles, Acta Math.146 (1981), 209–230
-,Rank Techniques and jump stratifications, In Proc. Bombay Conf. 1984, à paraître
-,Sur l'idéal de la partie finie générique de ℙ n, en préparation
Masser,D.,A note on multiplicities of polynomials, Publ. Math. ParisVI, n°43, Problème Diophantiens 1980/81, exp.n°2, 11pp.
Nagata,M.,On the fourteenth problem of Hilbert, Amer. J.Math.8l, n°3 (1959), 766–772
—,Lectures on the fourteenth problem of Hilbert, Tata Inst. of Fundamental Research, Bombay 1965
Perrin, D.,Courbes passant par k points généraux de ℙ 3, CRAS299, série I n°10 (1984), 451–453
Peskine,Ch.-Szpiro,Liaison des variétés algébriques I, Invent. Math.26 (1974) 271–302
Philippon, P.,Interpolation dans les espaces affines, In sem. de Théorie des Nombres 1980/81, Prog. in Math.22 (1982) 221–235
Skoda,H., Estimations L2 pour l'opérateur\(\bar \partial \) et applications arithmétiques, In sem. P.Lelong, L.N. in Math.578 (1977), 314–323
Waldschmidt M.,Propriétés arithmétiques de fonctions de plusieurs variables (II), In Sem. P.Lelong, L.N. in Math.578 (1977), 108–135
-,Nombres transcendants et groupes algébriques, Astérisque n°69–70 (1980)
Watanabe,M.,On projective Normality of Space Curves on a Non-Singular Cubic Surface in ℙ 3, Tokyo J.Math., vol.4 n°2 (1981), 331–341
Wüstholz,G.,On the degree of algebraic hypersurfaces with given singularities, Publ.Math. Paris VI, N°43, Problèmes Diophantiens 1980/8l, exp. n°5, 19pp.
Zariski,O.,Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order n and with a given number d of nodes, Amer. J.Math.103 (1982)
Author information
Authors and Affiliations
Additional information
En hommage au Professeur Karl STEIN
Rights and permissions
About this article
Cite this article
Hirschowitz, A. La Methode d1Horace pour l'Interpolation à Plusieurs Variables. Manuscripta Math 50, 337–388 (1985). https://doi.org/10.1007/BF01168836
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01168836