Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy

The multiplicity of a zero of a restriction of a polynomial of degree $q$ in ${\Bbb C}^n,\ n\ge 2$, to a non-singular trajectory of a polynomial vector field $\xi$ with coefficients of degree $p$ does not exceed $[pq(p+q)]^{2^{n-2}}$ if the polynomial does not vanish identically on this trajectory. For a system of polynomial vector fields, this implies an effective estimate on degree of nonholonomy, i.e., the minimal order of brackets necessary to generate a subspace of maximal possible dimension at each point. In particular, this allows one to check effectively whether a given system of polynomial vector fields is totally nonholonomic (controllable) at each point. Similar estimates are found for systems of vector fields with analytic coefficients satisfying polynomial Pfaffian equations. This allows one to check effectively whether such a system is totally nonholonomic (controllable) at a given point.

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Published 1 January 1995