Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

  • Y. YomdinEmail author
Original Paper Area 1


Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently rather separated domains: smooth dynamics, diophantine geometry, approximation theory, and computational geometry. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the paper. We present also some new results, stressing interconnection between various types and various applications of smooth parametrization.


Smooth parametrization Entropy Counting rational points  Remez-type inequalities 

Mathematics Subject Classification

37C05 11Gxx 41Axx 



The author would like to thank D. Burguet, G. Comte, O. Friedland, Y. Ishii, G. Jones, G. Liao, P. Milman, B. Mourrain, J. Pila, R. Pierzchava, M. Thomas, A. Wilkie for useful discussions, and for explaining him some topics presented below. Special thanks belong to RIMS Institute, Kyoto, and to the organizers of the conference there in July 2013 on semi-algebraic techniques in Dynamics, which inspired a good part of this paper.


  1. 1.
    Alberti, L., Mourrain, B., Tecourt, J.-P.: Isotopic triangulation of a real algebraic surface. J. Symb. Comput. 44(9), 1291–1310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baran, M., Pleśniak, W.: Bernstein and van der Corput–Shaake type inequalities on semialgebraic curves. Stud. Math. 125(1), 83–96 (1997)zbMATHGoogle Scholar
  3. 3.
    Baran, M., Pleśniak, W.: Polynomial inequalities on algebraic sets. Stud. Math. 141(3), 209–219 (2000)zbMATHGoogle Scholar
  4. 4.
    Baran, M., Pleśniak, W.: Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities. Stud. Math. 141(3), 221–234 (2000)zbMATHGoogle Scholar
  5. 5.
    Batenkov, D., Yomdin, Y.: Taylor Domination, Turán Lemma, and Poincaré–Perron Sequences. In: Nonlinear Analysis and Optimization. Contemporary Mathematics, AMS (to appear)Google Scholar
  6. 6.
    Batenkov, D., Yomdin, Y.: Geometry and singularities of the Prony mapping. J. Singul. 10, 1–25 (2014)MathSciNetGoogle Scholar
  7. 7.
    Benedetti, R., Risler, J.J.: Real algebraic and semi-algebraic sets. In: Actualites Mathematiques. Hermann, Paris (1990)Google Scholar
  8. 8.
    Bierstone, E., Milman, P.: Semianalytic and subanalytic sets. IHES Publ. Math. 67, 5–42 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bierstone, E., Grigoriev, D., Wlodarczyk, J.: Effective Hironaka resolution and its complexity. Asian J. Math. 15(2), 193–228 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bombieri, E., Pila, J.: The number of integral points on arcs and ovals. Duke Math. J. 59(2), 337–357 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bos, L., Levenberg, N., Milman, P., Taylor, B.A.: Tangential Markov inequalities characterize algebraic submanifolds of \(C^n\). Indiana Univ. Math. J. 44, 115–137 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bos, L., Levenberg, N., Milman, P., Taylor, B.A.: Tangential Markov inequalities on real algebraic varieties. Indiana Univ. Math. J. 47(4), 1257–1272 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bos, L.P., Brudnyi, A., Levenberg, N.: On polynomial inequalities on exponential curves in \(\mathbb{C}^n\). Constr. Approx. 31(1), 139–147 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bourgain, J., Goldstein, M., Schlag, W.: Anderson localization for Schrodinger operators on \(Z^2\) with quasi-periodic potential. Acta Math. 188, 41–86 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brudnyi, A.: On local behavior of holomorphic functions along complex submanifolds of \({\mathbb{C}}^N\). Invent. Math. 173(2), 315–363 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Brudnyi, A., Yomdin, Y.: Norming Sets and related Remez-type Inequalities (2013, preprint). arXiv:1312.6050
  17. 17.
    Burguet, D.: A proof of Yomdin–Gromov’s algebraic lemma. Isr. J. Math. 168, 291–316 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Burguet, D.: Quantitative Morse–Sard theorem via algebraic lemma. C. R. Math. Acad. Sci. Paris 349(7–8), 441–443 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Burguet, D.: Existence of measures of maximal entropy for \(C^r\) interval maps. Proc. Am. Math. Soc. 142(3), 957–968 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Burguet, D., Liao, G., Yang, J.: Asymptotic h-expansiveness rate of \(C^\infty \) maps (2014, preprint)Google Scholar
  21. 21.
    Butler, L.: Some cases of Wilkie’s conjecture. Bull. Lond. Math. Soc. 44(4), 642–660 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Cluckers, R., Comte, G., Loeser, F.: Non-archimedean Yomdin–Gromov parametrization and points of bounded height (preprint). arXiv:1404.1952v1
  23. 23.
    Coman, D., Poletsky, E.A.: Transcendence measures and algebraic growth of entire functions. Invent. Math. 170, 103–145 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Coman, D., Poletsky, E.A.: Polynomial estimates, exponential curves and Diophantine approximation. Math. Res. Lett. 17, 1125–1136 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    De Thelin, H., Vigny, G.: Entropy of meromorphic maps and dynamics of birational maps. Mem. Soc. Math. Fr. (N.S.) 122, vi+98 pp (2010)Google Scholar
  26. 26.
    Diatta, D.N., Mourrain, B., Ruatta, O.: On the isotopic meshing of an algebraic implicit surface. J. Symb. Comput. 47(8), 903–925 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Van den Dries, L.: Tame topology and O-minimal structures. In: London Mathematical Society Lecture Note Series, vol. 248. Cambridge University Press, Cambridge (1998)Google Scholar
  28. 28.
    Elihai, Y., Yomdin, Y.: Flexible high order discretization of geometric data for global motion planning, Theor. Comput. Sci. A 157, 53–77 (1996)Google Scholar
  29. 29.
    Fisher, A.: \(O\)-minimal, \(\Lambda ^m\)-regular stratification. Ann. Pure Appl. Log. 147, 101–112 (2007)CrossRefGoogle Scholar
  30. 30.
    Grigoriev, D., Milman, P.D.: Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2. Adv. Math. 231(6), 3389–3428 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Gromov, M.: Entropy, homology and semialgebraic geometry (after Y. Yomdin). Séminaire Bourbaki, vol. 1985/86Google Scholar
  32. 32.
    Gromov, M.: Spectral geometry of semi-algebraic sets. Ann. Inst. Fourier (Grenoble) 42(1–2), 249–274 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Guedj, V.: Entropie topologique des applications méromorphes. Ergod. Theory Dyn. Syst. 25, 1847–1855 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Haviv, D., Yomdin, Y.: Uniform approximation of near-singular surfaces. Theor. Comput. Sci. 392(1–3), 92–100 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hayman, W.K.: Multivalent Functions, 2nd edn. Cambridge University Press, Cambridge (1994)CrossRefzbMATHGoogle Scholar
  36. 36.
    Hironaka, H.: Triangulations of algebraic sets. Proc. Symp. Pure Math. Am. Math. Soc. 29, 165–185 (1975)CrossRefGoogle Scholar
  37. 37.
    Ishii, Y., Sands, D.: On some conjectures concerning the entropy of Lozi maps (2013, preprint)Google Scholar
  38. 38.
    Jones, G.O., Thomas, M.E.M.: The density of algebraic points on certain Pfaffian surfaces. Q. J. Math. 63, 637–651 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Jones, G.O., Miller, D.J., Thomas, M.E.M.: Mildness and the density of rational points on certain transcendental curves. Notre Dame J. Form. Log. 52(1), 67–74 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Liao, G.: Entropy of analytic maps (2012, preprint)Google Scholar
  41. 41.
    Liao, G., Viana, M., Yang, J.: The entropy conjecture for diffeomorphisms away from tangencies. J. Eur. Math. Soc. 15(6), 2043–2060 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    McMullen, C.: Entropy on Riemann surfaces and the Jacobians of finite covers. Comment. Math. Helv. 88(4), 953–964 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Marmon, O.: A generalization of the Bombieri–Pila determinant method. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), Issledovaniya po Teorii Chisel. 10, 63–77, 242 [Translation in J. Math. Sci. (N. Y.) 171(6), 736–744 (2010)]Google Scholar
  44. 44.
    Masser, D.: Rational values of the Riemann zeta function. J. Number Theory 131, 2037–2046 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Milnor, J.: Is entropy effectively computable?, in a site “Open Problems in Dynamics and Ergodic Theory”.
  46. 46.
    Moncet, A.: Real versus complex volumes on real algebraic surfaces. Int. Math. Res. Not. 2012(16), 3723–3762Google Scholar
  47. 47.
    Mourrain, B., Wintz, J.: A subdivision method for arrangement computation of semi-algebraic curves. In: Nonlinear Computational Geometry, pp. 165-187, The IMA Volumes in Mathematics and its Applications, vol. 151, Springer, New York (2010)Google Scholar
  48. 48.
    Narayan, K.L.: Computer Aided Design and Manufacturing. Prentice Hall of India, New Delhi (2008)Google Scholar
  49. 49.
    Newhouse, S.: Entropy and volume. Ergod. Theory Dyn. Syst. 8 \(^*\)(Charles Conley Memorial Issue), 283–299 (1988)Google Scholar
  50. 50.
    Newhouse, S.: Continuity properties of entropy. Ann. Math. (2) 129(2), 215–235 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Newhouse, S., Berz, M., Grote, J., Makino, K.: On the estimation of topological entropy on surfaces. In: Geometric and Probabilistic Structures in Dynamics, pp. 243–270. Contemporary Mathematics, vol. 469. American Mathematical Society, Providence (2008)Google Scholar
  52. 52.
    Nonlinear Computational Geometry. In: Emeris, I.Z., Theobald, Th., Sottile, F. (eds.) The IMA Volumes in Mathematics and its Applications, vol. 151. Springer, New York (2010)Google Scholar
  53. 53.
    Pierzchala, R.: Remez-type inequality on sets with cusps (2012, preprint)Google Scholar
  54. 54.
    Pierzchala, R.: Markov’s inequality in the o-minimal structure of convergent generalized power series. Adv. Geom. 12(4), 647–664 (2012)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Pierzchala, R.: UPC condition in polynomially bounded o-minimal structures. J. Approx. Theory 132(1), 25–33 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Pila, J.: Geometric postulation of a smooth function and the number of rational points. Duke Math. J. 63, 449–463 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Pila, J.: Geometric and arithmetic postulation of the exponential function. J. Aust. Math. Soc. Ser. A 54, 111–127 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Pila, J.: Integer points on the dilation of a subanalytic surface. Q. J. Math. 55(Part 2), 207–223 (2004)Google Scholar
  59. 59.
    Pila, J.: Rational points on a subanalytic surface. Annales De l’Institut Fourier, Grenoble 55(5), 1501–1516 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Pila, J.: Mild parametrization and the rational points on a Pfaff curve. Commentarii Mathematici Universitatis Sancti Pauli 55, 1–8 (2006)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Pila, J.: On the algebraic points of a definable set. Sel. Math. N. S. 15, 151–170 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Pila, J.: Counting rational points on a certain exponential-algebraic surface. Annales De l’Institut Fourier, Grenoble 60(2), 489–514 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Pila, J., Wilkie, A.J.: The rational points of a definable set. Duke Math. J. 133(3), 591–616 (2006)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Remez, E.J.: Sur une propriete des polynomes de Tchebycheff. Comm. Inst. Sci. Kharkov 13, 93–95 (1936)Google Scholar
  65. 65.
    Roytvarf, N., Yomdin, Y.: Bernstein classes. Annales De l’Institut Fourier, Grenoble 47(3), 825–858 (1997)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Scanlon, T.: Counting special points: logic, diophantine geometry, and transcendence theory. Bull. AMS 49(1), 51–71 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Scanlon, T.: A Euclidean Skolem–Mahler–Lech–Chabauty method. Math. Res. Lett. 18(5), 833–842 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Thomas, M.E.M.: An o-minimal structure without mild parameterization. Ann. Pure Appl. Log. 162(6), 409–418 (2011)CrossRefzbMATHGoogle Scholar
  69. 69.
    Thomas, M.E.M.: Convergence results for function spaces over o-minimal structures. J. Log. Anal. 4, Paper 1, 14 pp (2012)Google Scholar
  70. 70.
    Wilkie, A.J.: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Am. Math. Soc. 9, 1051–1094 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Wittig, A., Berz, M., Grote, J., Makino, K., Newhouse, S.: Rigorous and accurate enclosure of invariant manifolds on surfaces. Regul. Chaotic Dyn. 15(2–3), 107–126 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Xu, G., Mourrain, B., Duvigneau, R., Galligo, A.: Parameterization of computational domain in isogeometric analysis: methods and comparison. Comput. Methods Appl. Mech. Eng. 200(23–24), 2021–2031 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Xu, G., Mourrain, B., Duvigneau, R., Galligo, A.: Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications. Comput. Aided Des. 45(2), 395–404 (2013)MathSciNetCrossRefGoogle Scholar
  74. 74.
    Xu, G., Mourrain, B., Duvigneau, R., Galligo, A.: Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis. Comput. Aided Des. 45(4), 812–821 (2013)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Yomdin, Y.: Volume growth and entropy. Isr. J. Math. 57(3), 285–300 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Yomdin, Y.: \(C^k\)-resolution of semialgebraic sets and mappings. Isr. J. Math. 57(3), 301–317 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Yomdin, Y.: Local complexity growth for iterations of real analytic mappings and semi-continuity moduli of the entropy. Ergod. Theory Dyn. Syst. 11, 583–602 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Yomdin, Y.: Semialgebraic complexity of functions. J. Complex. 21(1), 111–148 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Yomdin, Y.: Some quantitative results in singularity theory. Ann. Polon. Math. 87, 277–299 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Yomdin, Y.: Generic singularities of surfaces, singularity theory. World Scientific Publishing, Hackensack (2007)Google Scholar
  81. 81.
    Yomdin, Y.: Analytic reparametrization of semialgebraic sets. J. Complex. 24(1), 54–76 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Yomdin, Y.: Remez-type inequality for discrete sets. Isr. J. Math. 186, 45–60 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Yomdin, Y.: Generalized Remez inequality for \((s, p)\)-valent functions (2013, preprint). arXiv:1102.2580
  84. 84.
    Yomdin, Y., Comte, G.: Tame geometry with application in smooth analysis. In: Lecture Notes in Mathematics, vol. 1834. Springer, Berlin (2004)Google Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations