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].