Multiplicity of polynomials on trajectories of polynomial vector fields in ${\displaystyle C^3}$

Let ξ be a polynomial vector field on ${\displaystyle ^n}$ 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 ${\displaystyle p+2p{(p+d-1)}^2}$. 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).