Multiplicity of Polynomials on Trajectories of Polynomials Vector Fields in C3

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).
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Submitted on : Friday, June 20, 2014 - 1:52:54 PM
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  • HAL Id : hal-01010760, version 1

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Andrei Gabrielov, Frédéric Jean, Jean-Jacques Risler. Multiplicity of Polynomials on Trajectories of Polynomials Vector Fields in C3. Banach Center Publications, 1998, 44 (1), pp.109-121. ⟨hal-01010760⟩

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