Multiplicity of Polynomials on Trajectories of Polynomials Vector Fields in C3

Andrei Gabrielov, Frédéric Jean and Jean-Jacques Risler
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
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},
}