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Smooth parametrizations in dynamics, analysis, diophantine and computational geometry. (English) Zbl 1358.37045
Summary: 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.
Reviewer: Reviewer (Berlin)

MSC:
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
03C64 Model theory of ordered structures; o-minimality
11G05 Elliptic curves over global fields
41A15 Spline approximation
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References:
[1] Alberti, L; Mourrain, B; Tecourt, J-P, Isotopic triangulation of a real algebraic surface, J. Symb. Comput., 44, 1291-1310, (2009) · Zbl 1173.14344
[2] Baran, M; Pleśniak, W, Bernstein and Van der Corput-shaake type inequalities on semialgebraic curves, Stud. Math., 125, 83-96, (1997) · Zbl 0895.41011
[3] Baran, M; Pleśniak, W, Polynomial inequalities on algebraic sets, Stud. Math., 141, 209-219, (2000) · Zbl 0987.41006
[4] Baran, M; Pleśniak, W, Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities, Stud. Math., 141, 221-234, (2000) · Zbl 0987.41005
[5] Batenkov, D., Yomdin, Y.: Taylor Domination, Turán Lemma, and Poincaré-Perron Sequences. In: Nonlinear Analysis and Optimization. Contemporary Mathematics, AMS (to appear) · Zbl 1359.30011
[6] Batenkov, D; Yomdin, Y, Geometry and singularities of the prony mapping, J. Singul., 10, 1-25, (2014) · Zbl 1353.94014
[7] Benedetti, R., Risler, J.J.: Real algebraic and semi-algebraic sets. In: Actualites Mathematiques. Hermann, Paris (1990) · Zbl 0694.14006
[8] Bierstone, E; Milman, P, Semianalytic and subanalytic sets, IHES Publ. Math., 67, 5-42, (1988) · Zbl 0674.32002
[9] Bierstone, E; Grigoriev, D; Wlodarczyk, J, Effective Hironaka resolution and its complexity, Asian J. Math., 15, 193-228, (2011) · Zbl 1315.14022
[10] Bombieri, E; Pila, J, The number of integral points on arcs and ovals, Duke Math. J., 59, 337-357, (1989) · Zbl 0718.11048
[11] Bos, L; Levenberg, N; Milman, P; Taylor, BA, Tangential Markov inequalities characterize algebraic submanifolds of \(C^n\), Indiana Univ. Math. J., 44, 115-137, (1995) · Zbl 0824.41015
[12] Bos, L; Levenberg, N; Milman, P; Taylor, BA, Tangential Markov inequalities on real algebraic varieties, Indiana Univ. Math. J., 47, 1257-1272, (1998) · Zbl 0938.32008
[13] Bos, LP; Brudnyi, A; Levenberg, N, On polynomial inequalities on exponential curves in \(\mathbb{C}^n\), Constr. Approx., 31, 139-147, (2010) · Zbl 1183.41009
[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) · Zbl 1022.47023
[15] Brudnyi, A, On local behavior of holomorphic functions along complex submanifolds of \({\mathbb{C}}^N\), Invent. Math., 173, 315-363, (2008) · Zbl 1149.32002
[16] Brudnyi, A., Yomdin, Y.: Norming Sets and related Remez-type Inequalities (2013, preprint). arXiv:1312.6050 · Zbl 1342.41016
[17] Burguet, D, A proof of yomdin-gromov’s algebraic lemma, Isr. J. Math., 168, 291-316, (2008) · Zbl 1169.14038
[18] Burguet, D, Quantitative Morse-sard theorem via algebraic lemma, C. R. Math. Acad. Sci. Paris, 349, 441-443, (2011) · Zbl 1218.58027
[19] Burguet, D, Existence of measures of maximal entropy for \(C^r\) interval maps, Proc. Am. Math. Soc., 142, 957-968, (2014) · Zbl 1290.37020
[20] Burguet, D., Liao, G., Yang, J.: Asymptotic h-expansiveness rate of \(C^∞ \) maps (2014, preprint) · Zbl 1352.37015
[21] Butler, L, Some cases of wilkie’s conjecture, Bull. Lond. Math. Soc., 44, 642-660, (2012) · Zbl 1253.03063
[22] Cluckers, R., Comte, G., Loeser, F.: Non-archimedean Yomdin-Gromov parametrization and points of bounded height (preprint). arXiv:1404.1952v1 · Zbl 1393.11032
[23] Coman, D; Poletsky, EA, Transcendence measures and algebraic growth of entire functions, Invent. Math., 170, 103-145, (2007) · Zbl 1134.30018
[24] Coman, D; Poletsky, EA, Polynomial estimates, exponential curves and Diophantine approximation, Math. Res. Lett., 17, 1125-1136, (2010) · Zbl 1227.32005
[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) · Zbl 1214.37004
[26] Diatta, DN; Mourrain, B; Ruatta, O, On the isotopic meshing of an algebraic implicit surface, J. Symb. Comput., 47, 903-925, (2012) · Zbl 1237.14069
[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) · Zbl 0953.03045
[28] Elihai, Y., Yomdin, Y.: Flexible high order discretization of geometric data for global motion planning, Theor. Comput. Sci. A 157, 53-77 (1996) · Zbl 0871.68167
[29] Fisher, A, \(O\)-minimal, \(Λ ^m\)-regular stratification, Ann. Pure Appl. Log., 147, 101-112, (2007) · Zbl 1125.03029
[30] Grigoriev, D; Milman, PD, Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2, Adv. Math., 231, 3389-3428, (2012) · Zbl 1287.14004
[31] Gromov, M.: Entropy, homology and semialgebraic geometry (after Y. Yomdin). Séminaire Bourbaki, vol. 1985/86 · Zbl 0611.58041
[32] Gromov, M, Spectral geometry of semi-algebraic sets, Ann. Inst. Fourier (Grenoble), 42, 249-274, (1992) · Zbl 0759.58048
[33] Guedj, V, Entropie topologique des applications méromorphes, Ergod. Theory Dyn. Syst., 25, 1847-1855, (2005) · Zbl 1087.37015
[34] Haviv, D; Yomdin, Y, Uniform approximation of near-singular surfaces, Theor. Comput. Sci., 392, 92-100, (2008) · Zbl 1134.68061
[35] Hayman, W.K.: Multivalent Functions, 2nd edn. Cambridge University Press, Cambridge (1994) · Zbl 0904.30001
[36] Hironaka, H, Triangulations of algebraic sets, Proc. Symp. Pure Math. Am. Math. Soc., 29, 165-185, (1975)
[37] Ishii, Y., Sands, D.: On some conjectures concerning the entropy of Lozi maps (2013, preprint) · Zbl 0763.11025
[38] Jones, GO; Thomas, MEM, The density of algebraic points on certain Pfaffian surfaces, Q. J. Math., 63, 637-651, (2012) · Zbl 1253.03065
[39] Jones, GO; Miller, DJ; Thomas, MEM, Mildness and the density of rational points on certain transcendental curves, Notre Dame J. Form. Log., 52, 67-74, (2011) · Zbl 1220.03034
[40] Liao, G.: Entropy of analytic maps (2012, preprint) · Zbl 1210.11074
[41] Liao, G; Viana, M; Yang, J, The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc., 15, 2043-2060, (2013) · Zbl 1325.37031
[42] McMullen, C, Entropy on Riemann surfaces and the Jacobians of finite covers, Comment. Math. Helv., 88, 953-964, (2013) · Zbl 1285.32006
[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)] · Zbl 1288.11030
[44] Masser, D, Rational values of the Riemann zeta function, J. Number Theory, 131, 2037-2046, (2011) · Zbl 1267.11091
[45] Milnor, J.: Is entropy effectively computable?, in a site “Open Problems in Dynamics and Ergodic Theory”. http://iml.univ-mrs.fr/kolyada/opds/ · Zbl 1290.37020
[46] Moncet, A.: Real versus complex volumes on real algebraic surfaces. Int. Math. Res. Not. 2012(16), 3723-3762 · Zbl 1252.14039
[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) · Zbl 1191.68890
[48] Narayan, K.L.: Computer Aided Design and Manufacturing. Prentice Hall of India, New Delhi (2008)
[49] Newhouse, S.: Entropy and volume. Ergod. Theory Dyn. Syst. 8\(^*\)(Charles Conley Memorial Issue), 283-299 (1988) · Zbl 0638.58016
[50] Newhouse, S, Continuity properties of entropy, Ann. Math. (2), 129, 215-235, (1989) · Zbl 0676.58039
[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) · Zbl 1154.37009
[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) · Zbl 1283.37078
[53] Pierzchala, R.: Remez-type inequality on sets with cusps (2012, preprint) · Zbl 1373.41008
[54] Pierzchala, R, Markov’s inequality in the o-minimal structure of convergent generalized power series, Adv. Geom., 12, 647-664, (2012) · Zbl 1271.32014
[55] Pierzchala, R, UPC condition in polynomially bounded o-minimal structures, J. Approx. Theory, 132, 25-33, (2005) · Zbl 1073.32002
[56] Pila, J, Geometric postulation of a smooth function and the number of rational points, Duke Math. J., 63, 449-463, (1991) · Zbl 0763.11025
[57] Pila, J, Geometric and arithmetic postulation of the exponential function, J. Aust. Math. Soc. Ser. A, 54, 111-127, (1993) · Zbl 0773.11046
[58] Pila, J.: Integer points on the dilation of a subanalytic surface. Q. J. Math. 55(Part 2), 207-223 (2004) · Zbl 1111.32004
[59] Pila, J, Rational points on a subanalytic surface, Annales De l’Institut Fourier, Grenoble, 55, 1501-1516, (2005) · Zbl 1121.11032
[60] Pila, J, Mild parametrization and the rational points on a Pfaff curve, Commentarii Mathematici Universitatis Sancti Pauli, 55, 1-8, (2006) · Zbl 1129.11029
[61] Pila, J, On the algebraic points of a definable set, Sel. Math. N. S., 15, 151-170, (2009) · Zbl 1218.11068
[62] Pila, J, Counting rational points on a certain exponential-algebraic surface, Annales De l’Institut Fourier, Grenoble, 60, 489-514, (2010) · Zbl 1210.11074
[63] Pila, J; Wilkie, AJ, The rational points of a definable set, Duke Math. J., 133, 591-616, (2006) · Zbl 1217.11066
[64] Remez, EJ, Sur une propriete des polynomes de Tchebycheff, Comm. Inst. Sci. Kharkov, 13, 93-95, (1936)
[65] Roytvarf, N; Yomdin, Y, Bernstein classes, Annales De l’Institut Fourier, Grenoble, 47, 825-858, (1997) · Zbl 0974.30524
[66] Scanlon, T, Counting special points: logic, Diophantine geometry, and transcendence theory, Bull. AMS, 49, 51-71, (2012) · Zbl 1323.11041
[67] Scanlon, T, A Euclidean Skolem-Mahler-Lech-Chabauty method, Math. Res. Lett., 18, 833-842, (2011) · Zbl 1283.37078
[68] Thomas, MEM, An o-minimal structure without mild parameterization, Ann. Pure Appl. Log., 162, 409-418, (2011) · Zbl 1251.03043
[69] Thomas, M.E.M.: Convergence results for function spaces over o-minimal structures. J. Log. Anal. 4, Paper 1, 14 pp (2012)
[70] Wilkie, AJ, 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) · Zbl 0892.03013
[71] Wittig, A; Berz, M; Grote, J; Makino, K; Newhouse, S, Rigorous and accurate enclosure of invariant manifolds on surfaces, Regul. Chaotic Dyn., 15, 107-126, (2010) · Zbl 1203.37030
[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, 2021-2031, (2011) · Zbl 1228.65232
[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, 395-404, (2013)
[74] Xu, G; Mourrain, B; Duvigneau, R; Galligo, A, Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis, Comput. Aided Des., 45, 812-821, (2013)
[75] Yomdin, Y, Volume growth and entropy, Isr. J. Math., 57, 285-300, (1987) · Zbl 0641.54036
[76] Yomdin, Y, \(C^k\)-resolution of semialgebraic sets and mappings, Isr. J. Math., 57, 301-317, (1987) · Zbl 0641.54037
[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) · Zbl 0756.58041
[78] Yomdin, Y, Semialgebraic complexity of functions, J. Complex., 21, 111-148, (2005) · Zbl 1101.68614
[79] Yomdin, Y, Some quantitative results in singularity theory, Ann. Polon. Math., 87, 277-299, (2005) · Zbl 1090.58021
[80] Yomdin, Y.: Generic singularities of surfaces, singularity theory. World Scientific Publishing, Hackensack (2007) · Zbl 1125.58011
[81] Yomdin, Y, Analytic reparametrization of semialgebraic sets, J. Complex., 24, 54-76, (2008) · Zbl 1143.32007
[82] Yomdin, Y, Remez-type inequality for discrete sets, Isr. J. Math., 186, 45-60, (2011) · Zbl 1258.41008
[83] Yomdin, Y.: Generalized Remez inequality for \((s, p)\)-valent functions (2013, preprint). arXiv:1102.2580
[84] Yomdin, Y., Comte, G.: Tame geometry with application in smooth analysis. In: Lecture Notes in Mathematics, vol. 1834. Springer, Berlin (2004) · Zbl 1076.14079
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