| Título: | Computation of topological numbers via linear algebra: hypersurfaces, vector fields and vector fields on hypersurfaces. |
| Autores: | Giraldo Suárez, Luis ; Gómez-Mont, X. ; Mardešic, P. |
| Tipo de documento: | texto impreso |
| Editorial: | American Mathematical Society, 1999 |
| Palabras clave: | Estado = Publicado , Materia = Ciencias: Matemáticas: Geometria algebraica , Tipo = Sección de libro |
| Resumen: |
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 (?) by V R + for small positive real ? . 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 ? gives ?(V R + )=1?? . 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. |
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