%0 Book Section
%T Computation of topological numbers via linear algebra: hypersurfaces, vector fields and vector fields on hypersurfaces.
publisher American Mathematical Society
%D 1999
%U 0-8218-1381-1
%@ https://hdl.handle.net/20.500.14352/60710
%X 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.
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