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

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

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## Abstract

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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 $p+2p{\left(p+d-1\right)}^{2}$. 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).

## How to cite

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Gabrielov, 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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2. [2] A. Bellaïche, The tangent space in sub-Riemannian geometry, in: Sub-Riemannian Geometry, A. Bellaïche and J.-J. Risler (ed.), Progr. Math. 144, Birkhäuser, Basel, 1996, 1-78. Zbl0862.53031
3. [3] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1984. Zbl0541.14005
4. [4] A. Gabrielov, J.-M. Lion and R. Moussu, Ordre de contact de courbes intégrales du plan, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 219-221.
5. [5] A. Gabrielov, Multiplicities of zeroes of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, Math. Res. Lett. 2 (1995), 437-451. Zbl0845.32003
6. [6] A. Gabrielov, Multiplicities of Pfaffian intersections and the Łojasiewicz inequality, Selecta Math. (N. S.) 1 (1995), 113-127. Zbl0889.32005
7. [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. [8] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985. Zbl0563.13001
9. [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. [10] J.-J. Risler, A bound for the degree of nonholonomy in the plane, Theoret. Comput. Sci. 157 (1996), 129-136. Zbl0871.93024
11. [11] P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Ergeb. Math. Grenzgeb. 4, Springer, Berlin, 1967. Zbl0146.16901

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