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2 results on '"LOCUS (Mathematics)"'

1. Résolution à l'infini et courbes polaires affines

Author

Ivanovski, Dimce

Subjects
*CURVES, *POLYNOMIALS, *MATHEMATICAL variables, *LINEAR statistical models, *LOCUS (Mathematics)
Abstract

Abstract: Let f be a polynomial map of two variables and let l be a linear form. The affine polar curve of f, for the direction l, is the singular locus of the map . We introduce two resolutions at infinity: the small resolution Π and the total resolution . Let be the line at infinity. The task of this Note is to determine the intersection locus of the strict transform of the polar curve with and . To cite this article: D. Ivanovski, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Published
2006
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2. Quelques résultats sur certaines fonctions à lieu singulier de dimension 1

Author

Daniel Barlet, Barlet, Daniel, Brasselet, J.-P., Soares Ruas, M.A., Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Institut Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), Brasselet, J.-P., Soares Ruas, and M.A.

Subjects
Essential singularity, Pure mathematics, formal microlocal operators, 010102 general mathematics, Mathematical analysis, Hypersurface singularity, Holomorphic function, (a, 1 dimensional singular locus, [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], Isolated singularity, Adjunction, 01 natural sciences, Milnor number, Mathematics::Algebraic Geometry, b)-module, 0103 physical sciences, Torsion (algebra), Brieskorn module, [MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV], 010307 mathematical physics, 0101 mathematics, Locus (mathematics), Mathematics, 32S05, 32S25, 32S40
Abstract

This text is a survey on my recent work [B.04] on some holomorphic germs having a one dimensional singular locus. An analogous of the Brieskorn module of an isolated singularity is defined and a finiteness theorem is proved using Kashiwara’s constructibility theorem. A bound for the (finite dimensional) torsion is also obtained. Non existence of torsion is proved for curves (reduced or not) an this property is stable by “Thom-Sebastiani” adjunction of an isolated singularity. This provides a lot of examples in any dimension where our formula r = μ(f)+v(f) generalizing the Milnor number formula, is valid.

Published
2006

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