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Multiplicity of a zero of an analytic function on a trajectory of a vector field. (English) Zbl 0948.32010
Bierstone, Edward (ed.) et al., The Arnoldfest. Proceedings of a conference in honour of V. I. Arnold for his 60th birthday, Toronto, Canada, June 15-21, 1997. Providence, RI: American Mathematical Society. Fields Inst. Commun. 24, 191-200 (1999).
Summary: The multiplicity $$\mu$$ of a zero of a restriction of an analytic function $$P$$ in $$\mathbb{C}^n$$ to a trajectory of a vector field $$\xi$$ with analytic coefficients is equal to the sum of the Euler characteristics of Milnor fibers associated with a deformation of $$P$$. When $$P$$ is a polynomial of degree $$p$$ and $$\xi$$ is a vector field with polynomial coefficients of degree $$q$$, this allows one to compute $$\mu$$ in purely algebraic terms, and to give an upper bound for $$\mu$$ in terms of $$n, p, q$$, single exponential in $$n$$ and polynomial in $$p, q$$. This implies a single exponential in $$n$$ bound on degree of nonholonomy of a system of polynomial vector fields in $$\mathbb{C}^n$$.
For the entire collection see [Zbl 0929.00102].

##### MSC:
 32B20 Semi-analytic sets, subanalytic sets, and generalizations 14P05 Real algebraic sets