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The index of a vector field tangent to an odd-dimensional hypersurface and the signature of the relative Hessian. (English. Russian original) Zbl 0946.32016
Funct. Anal. Appl. 33, No. 1, 1-10 (1999); translation from Funkts. Anal. Prilozh. 33, No. 1, 1-13 (1999).
The paper gives a formula for the (Poincaré-Hopf) index $$\text{Ind}_{\gamma, \pm,0}(x)$$ (introduced by X. Gómez-Mont, J. Seade and A. Verjovsky in Math. Ann. 291, 737-751 (1991; Zbl 0725.32012)) of a vector field $$X$$ tangent to a hypersurface $$V=f^{-1}(0) \subset B\subset \mathbb{R}^{n+1}$$ where $$f$$ is a real-analytic function defined on the unit ball $$B$$ of $$\mathbb{R}^{n+1}$$ with an algebraic isolated singularity at 0.

MSC:
 32S65 Singularities of holomorphic vector fields and foliations 32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants 32S25 Complex surface and hypersurface singularities
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References:
 [1] V. Arnold, ”The index of a singular point of a vector field, the Petrovskii-Oleinik inequality and mixed Hodge structures,” Funkts. Anal. Prilozhen.,12, No. 1, 1–14 (1978). [2] V. Arnold, S. Gusein-Zade, and A. Varchenko, Singularities of Differentiable Maps, I, Birkhauser, 1985. [3] Ch. Bonatti and X. Gómez-Mont, ”The index of a holomorphic vector field on a singular variety, I,” Asterisque,222, 9–35 (1994). · Zbl 0810.32017 [4] W. Ebeling and S. M. Gusein-Zade, On the Index of a Vector Field at an Isolated Singularity, Preprint. [5] D. Eisenbud and H. Levine, ”An algebraic formula for the degree of aC map germ,” Ann. Math.,106, 19–38 (1977). · Zbl 0398.57020 [6] X. Gómez-Mont, An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, Preprint Communicaciones del CIMAT I-96-27, 1996, pp. 1–23. [7] X. Gómez-Mont and P. Mardešić, ”The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant,” Ann. Inst. Fourier (Grenoble),47, No. 5 (1997). · Zbl 0891.32013 [8] X. Gómez-Mont J. Seade, and A. Verjovsky, ”The index of a holomorphic flow with an isolated singularity,” Math. Ann.291, 737–751 (1991). · Zbl 0725.32012 [9] H. Grauert and H. Remmert, Coherent Analytic Sheaves, Grundlehren 265, Springer-Verlag, 1984. · Zbl 0537.32001 [10] G. N. Khimshiashvili, ”On the local degree of a smooth map,” Trudy Tbilisi Math. Inst., 105–124 (1980). · Zbl 0526.58010
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