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1998 | 44 | 1 | 109-121
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Multiplicity of polynomials on trajectories of polynomial vector fields in $C^3$

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Let ξ be a polynomial vector field on $𝐂^n$ 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 $p + 2p(p+d-1)^2$. 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).
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  • Department of Mathematics, Purdue University, W. Lafayette, IN 47907-1395
  • École Nationale Supérieure de Techniques Avancées, 32, Bd Victor, F-75739 Paris cedex 15
  • Équipe Analyse Algébrique, Institut de Mathématiques, Université Paris 6, Case 82, 2 place Jussieu, F-75252 Paris Cedex 05
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