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The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant. (English) Zbl 0891.32013
Let \(\delta= \sum^n_{i=0} h_i{\partial \over \partial x_i}\) be a real analytic vector field on \(\mathbb{R}^{n+1}\) with an algebraically isolated singularity at 0 and \(B= A_{\mathbb{R}^{n +1}, 0}/(h_0, \dots, h_n)\), \(A_{\mathbb{R}^{n+1}, 0}\), the ring of germs of real analytic functions on \(\mathbb{R}^{n+1}\) at 0.
The product in the algebra \(B\) and a linear map \(\ell: B\to \mathbb{R}\) with \(\ell(\text{det} ({\partial h_i \over \partial x_j})) >0\) defines a non-degenerate bilinear form \(\langle,\rangle: B\times B\to \mathbb{R}\). The index of \(\delta\) is the signature of this bilinear form.
Let \(V\subseteq \mathbb{R}^{n+1}\) be a hypersurface with an isolated singularity at 0 defined by \(f=0\) such that \(\delta\) is tangent to \(V\), i.e. \(\delta(f) \in(f)\).
Let \(h= {\delta(f) \over f}\) and define \(J_f(\delta) ={\text{det} ({\partial h_i \over \partial x_j}) \over h} \in B/ \text{Ann} (h)\) the relative Jacobian determinant of \(\delta\). Choose a linear map \(\ell:B/ \text{Ann} (h)\to \mathbb{R}\) such that \(\ell(J_f (\delta)) >0\).
The product in \(B/\text{Ann} (h)\) together with \(\ell\) defines a binlinear form on \(B/ \text{Ann} (h)\), let \(\text{Sgn}_{f,0} (\delta)\) be the signature of this bilinear form.
It is proved that the function \(\text{Sgn}_{f,0}\) satisfies the law of conservation of number: \[ \text{Sgn}_{f,0} (\delta)= \text{Sgn}_{f,0} (\delta_t) +\sum_{x\in V \setminus \{0\}\atop \delta_t(x) =0}\text{Index}_{V,x} (\delta_t|V) \] for \(x\) close to 0 and \(\delta_t\) tangent to \(V\) and close to \(\delta\).

MSC:
32S05 Local complex singularities
32S25 Complex surface and hypersurface singularities
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
37G99 Local and nonlocal bifurcation theory for dynamical systems
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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