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Multiplicity of polynomials on trajectories of polynomial vector fields in $C^3$

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Gabrielov A., Jean F., Risler J.-J.

Banach Center Publications

1998

tom 44

nr 1

109-121

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).

Ograniczanie wyników

1 Banach Center Publications

1 Gabrielov A.

1 Jean F.

1 Risler J.-J.

1 1998

Wyniki wyszukiwania

Multiplicity of polynomials on trajectories of polynomial vector fields in $C^3$