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Giraldo Suárez, Luis and Gómez-Mont, X. and Mardešic, P.
(1999)
*Computation of topological numbers via linear algebra: hypersurfaces, vector fields and vector fields on hypersurfaces.*
In
Complex geometry of groups.
Contemporary Mathematics
(240).
American Mathematical Society, Providence, pp. 175-182.
ISBN 0-8218-1381-1

Official URL: http://www.ams.org/books/conm/240/

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## Abstract

In this paper the authors review some work relating the topology to algebraic invariants. Three cases are considered: hypersurfaces, vector fields and vector fields tangent to hypersurfaces.

An example is the case of "real Milnor fibres". Let f R :B R →R be a real analytic function which extends to a function f on the closed unit ball B in C n. Assume that 0 is the only critical point of f and denote the real hypersurface f −1 R (δ) by V R + for small positive real δ .

Let A denote A/(f 0 ,⋯,f n ) , the quotient of the ring of germs of real analytic functions by the partial derivatives of f . Choose any R-linear map L:A→R which sends the Hessian of f to a positive number; then one has a non-degenerate bilinear form ⟨ , ⟩:A×A→ . A→ L R , for which the signature σ gives χ(V R + )=1−σ .

The authors use a result of D. Eisenbud and H. I. Levine [Ann. Math. (2) 106 (1977), no. 1, 19–44, to calculate the signature of the bilinear form; this is notoriously difficult to compute in practice.

Item Type: | Book Section |
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Additional Information: | Proceedings of the 1st Iberoamerican "Cruz del Sur" Congress on Geometry held in Olmué, Chile, January 5–11, 1998 |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 21756 |

Deposited On: | 11 Jun 2013 12:57 |

Last Modified: | 12 Dec 2018 15:13 |

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