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Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy. (English) Zbl 0845.32003
The author gives the following effective estimates of the multiplicity \(\kappa\) of a zero of an arbitrary polynomial of degree \(q\) on a trajectory of a polynomial vector field with coefficients of degree \(p\) in \(\mathbb{C}^n\):
\[ \kappa \leq p^{2^{n - 2 - 1}} q^{ 2^{n - 2}} (p + q)^{2^{n - 2}}. \] The above estimate allows to define an effective upper bound on the degree of nonholonomy for a system of polynomial vector fields of degree \(p\) in \(\mathbb{C}^n\). The degree of nonholonomy plays an important role in the theory of several complex variables. Also totally nonholonomic systems appear in control theory, in the theory of hypoelliptic partial differential equations, in the probability theory as well as in differential geometry.

32A15 Entire functions of several complex variables
70F25 Nonholonomic systems related to the dynamics of a system of particles
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