# Multiplicity of Polynomials on Trajectories of Polynomials Vector Fields in C3

1998

Publication type:

Paper in peer-reviewed journals

Journal:

Banach Center Publications, vol. 44(1), pp. 109-121

Workshop:

Singularities Symposium - Lojasiewicz 70

Publisher:

Institute of Mathematics Polish Academy of Sciences

External link:

HAL:

Abstract:

Let ξ be a polynomial vector field on n with coefficients of degree d and P be a polynomial of degree p. We are interested in bounding the multiplicity of a zero of a restriction of P to a non-singular trajectory of ξ, when P does not vanish identically on this trajectory. Bounds doubly exponential in terms of n are already known ([9,5,10]). In this paper, we prove that, when n=3, there is a bound of the form p + 2 p ( p + d - 1 ) 2 . In Control Theory, such a bound can be used to give an estimate of the degree of nonholonomy for a system of polynomial vector fields (this degree expresses the level of Lie-bracketing needed to generate the tangent space at each point).

BibTeX:

@article{Gab-Jea-Ris-1998, author={Andrei Gabrielov and Frédéric Jean and Jean-Jacques Risler }, title={Multiplicity of Polynomials on Trajectories of Polynomials Vector Fields in C3 }, journal={Banach Center Publications }, year={1998 }, volume={44(1) }, pages={109--121}, }