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1998 | 44 | 1 | 109-121
Tytuł artykułu

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

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Języki publikacji
EN
Abstrakty
EN
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).
Słowa kluczowe
Rocznik
Tom
44
Numer
1
Strony
109-121
Opis fizyczny
Daty
wydano
1998
Twórcy
  • 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
Bibliografia
  • [1] V. I. Arnol$'$d, S. M. Guseĭn-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Birkhäuser, Boston, 1985.
  • [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.
  • [3] W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1984.
  • [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] 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.
  • [6] A. Gabrielov, Multiplicities of Pfaffian intersections and the Łojasiewicz inequality, Selecta Math. (N. S.) 1 (1995), 113-127.
  • [7] A. Gabrielov, Multiplicity of a Zero of an Analytic Function on a Trajectory of a Vector Field, Preprint, Purdue University, March 1997.
  • [8] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985.
  • [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.
  • [11] P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Ergeb. Math. Grenzgeb. 4, Springer, Berlin, 1967.
Typ dokumentu
Bibliografia
Identyfikatory
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bwmeta1.element.bwnjournal-article-bcpv44i1p109bwm
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