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

Andrei Gabrielov; Frédéric Jean; Jean-Jacques Risler

Banach Center Publications (1998)

- Volume: 44, Issue: 1, page 109-121
- ISSN: 0137-6934

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topGabrielov, Andrei, Jean, Frédéric, and Risler, Jean-Jacques. "Multiplicity of polynomials on trajectories of polynomial vector fields in $C^3$." Banach Center Publications 44.1 (1998): 109-121. <http://eudml.org/doc/208871>.

@article{Gabrielov1998,

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 $p + 2p(p+d-1)^2$. 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).},

author = {Gabrielov, Andrei, Jean, Frédéric, Risler, Jean-Jacques},

journal = {Banach Center Publications},

keywords = {multiplicity bounds; polynomial vector fields},

language = {eng},

number = {1},

pages = {109-121},

title = {Multiplicity of polynomials on trajectories of polynomial vector fields in $C^3$},

url = {http://eudml.org/doc/208871},

volume = {44},

year = {1998},

}

TY - JOUR

AU - Gabrielov, Andrei

AU - Jean, Frédéric

AU - Risler, Jean-Jacques

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

JO - Banach Center Publications

PY - 1998

VL - 44

IS - 1

SP - 109

EP - 121

AB - 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 $p + 2p(p+d-1)^2$. 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).

LA - eng

KW - multiplicity bounds; polynomial vector fields

UR - http://eudml.org/doc/208871

ER -

## References

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- [7] A. Gabrielov, Multiplicity of a Zero of an Analytic Function on a Trajectory of a Vector Field, Preprint, Purdue University, March 1997. Zbl0948.32010
- [8] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985. Zbl0563.13001
- [9] Y. V. Nesterenko, Estimates for the number of zeros of certain functions, in: New Advances in Transcendence Theory, A. Baker (ed.), Cambridge Univ. Press, Cambridge, 1988, 263-269.
- [10] J.-J. Risler, A bound for the degree of nonholonomy in the plane, Theoret. Comput. Sci. 157 (1996), 129-136. Zbl0871.93024
- [11] P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Ergeb. Math. Grenzgeb. 4, Springer, Berlin, 1967. Zbl0146.16901

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