Trajectories of polynomial vector fields and ascending chains of polynomial ideals
Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 563-609.

Nous donnons une borne supérieure complètement explicite pour le nombre d’intersections isolées entre une courbe intégrale d’un champ de vecteurs polynomial et une hypersurface algébrique dans l’espace euclidien de dimension quelconque. La borne est polynomiale par rapport à la hauteur des polynômes et la taille de la courbe, l’exposant étant une fonction explicite dépendant seulement du degré et de la dimension.

Le problème est alors très étroitement lié au problème de la longueur des chaînes ascendantes des idéaux polynomiaux, engendrées par les dérivations successives.

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in n and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension.

The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives.

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Novikov, Dmitri; Yakovenko, Sergei. Trajectories of polynomial vector fields and ascending chains of polynomial ideals. Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 563-609. doi : 10.5802/aif.1683. http://archive.numdam.org/articles/10.5802/aif.1683/

[1] V. I. Arnol’D, S. M. Guseǐn-Zade, A. N. Varchenko, Singularities of differentiable maps, Vol. I (The classification of critical points, caustics and wave fronts). Monographs in Mathematics, 82. Birkhäuser, Boston, Mass., 1985. | Zbl

[2] Miriam Briskin, Y. Yomdin, Algebraic families of analytic functions (I), J. Differential Equations 136, no. 2 (1997), 248-267. | MR | Zbl

[3] P. Enflo, V. Gurarii, V. Lomonosov, Yu. Lyubich, Exponential numbers of operators in normed spaces, Linear Algebra Appl., 219 (1995), 225-260. | MR | Zbl

[4] A. Gabrièlov, Multiplicities of zeros of polynomials on trajectories of polynomial vector fields and bounds on degree of nonholonomy, Math. Research Letters, 2 (1996), 437-451. | MR | Zbl

[5] A. Gabrièlov, Multiplicity of a zero of an analytic function on a trajectory of a vector field, Preprint, Purdue University, 1997, 7 pp.

[6] A. Gabrièlov, J.-M. Lion, R. Moussu, Ordre de contact de courbes intégrales du plan, C. R. Acad. Sci. Paris, Sér. I Math., 319, no. 3 (1994), 219-221. | MR | Zbl

[7] L. Gavrilov, Petrov modules and zeros of Abelian integrals. Preprint no. 95, Université Paul Sabatier (1997), to appear in Bull. Sci. Mathématiques. | Zbl

[8] Patrizia Gianni, B. Trager, G. Zacharias, Gröbner bases and primary decomposition of polynomial ideals. Computational aspects of commutative algebra, J. Symbolic Comput., 6, no. 2-3 (1988), 149-167. | Zbl

[9] M. Giusti, Some effectivity problems in polynomial ideal theory. EUROSAM 84 (Cambridge, 1984), 159-171, Lecture Notes in Comput. Sci., 174, Springer, Berlin-New York, 1984. | MR | Zbl

[10] A. Givental, Sturm's theorem for hyperelliptic integrals, Algebra i Analiz 1 (1989), no. 5, 95-102; translation in Leningrad Math. J., 1, no. 5 (1990), 1157-1163. | MR | Zbl

[11] J. Heintz, Definability and fast quantifier elimination in algebraically closed fields, Theoret. Comput. Sci., 24, no. 3 (1983), 239-277. | MR | Zbl

[12] Greta Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomialideale, Mathematische Annalen, 95 (1926), 736-788. | JFM

[13] W. V. D. Hodge, D. Pedoe, Methods of algebraic geometry. Vol. I. Reprint of the 1947 original. Cambridge University Press, Cambridge, 1994. | MR | Zbl

[14] Yu. Il’Yashenko, S. Yakovenko, Counting real zeros of analytic functions satisfying linear ordinary differential equations, J. Diff. Equations, 126, no. 1 (1996), 87-105. | MR | Zbl

[15] W. J. Kim, The Schwarzian derivative and multivalence, Pacific J. of Math., 31, no. 3 (1969), 717-724. | MR | Zbl

[16] Teresa Krick, A. Logar, An algorithm for the computation of the radical of an ideal in the ring of polynomials, Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA, 1991), 195-205, Lecture Notes in Comput. Sci., 539, Springer, Berlin, 1991. | MR | Zbl

[17] D. Lazard, A note on upper bounds for ideal-theoretic problems, J. Symbolic Comput., 13, no. 3 (1992), 231-233. | MR | Zbl

[18] S. Lojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel-Boston-Berlin, 1991. | MR | Zbl

[19] E. W. Mayr, A. R. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in Math., 46 (1982), 305-329. | MR | Zbl

[20] G. Moreno Socías, Length of polynomial ascending chains and primitive recursiveness, Math. Scand. 71 (1992), no. 2, 181-205; An Ackermannian polynomial ideal, in Applied algebra, algebraic algorithms and error-correcting codes (New Orleans, LA, 1991), 269-280, Lecture Notes in Comput. Sci., 539, Springer, Berlin, 1991; Autour de la fonction de Hilbert-Samuel (escaliers d'idéaux polynomiaux), Ph. D. Thesis, Centre de Mathématiques de l'École Polytechnique, 1991. | MR | Zbl

[21] D. Novikov, S. Yakovenko, Integral Frenet curvatures and oscillation of spatial curves around affine subspaces of a Euclidean space, J. of Dynamical and Control Systems, 2, no. 2 (1996), 157-191. | MR | Zbl

[22] Novikov, S. Yakovenko, Meandering of trajectories of polynomial vector fields in the affine n-space, Publ. Mat., 41, no. 1 (1997), 223-242. | MR | Zbl

[23] J.-J. Risler, A bound for the degree of nonholonomy in the plane, Algorithmic complexity of algebraic and geometric models (Creteil, 1994), Theoret. Comput. Sci., 157, no. 1 (1996), 129-136. | MR | Zbl

[24] A. Seidenberg, Constructions in algebra, Trans. Amer. Math. Soc., 197 (1974), 273-313. | MR | Zbl

[25] A. Seidenberg, Constructive proof of Hilbert's theorem on ascending chains, Trans. Amer. Math. Soc. 174 (1972), 305-312; On the length of a Hilbert ascending chain, Proc. Amer. Math. Soc., 29 (1971), 443-450. | Zbl

[26] Y. Yomdin, Oscillation of analytic curves, Proc. Amer. Math. Soc., 126, no. 2 (1998), 357-364. | MR | Zbl

[27] O. Zariski, P. Samuel, Commutative Algebra, vol. 1, Springer-Verlag, N. Y. et al., 1975, corrected reprinting of the 1958 edition.

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