Download PDF - Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalitiesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities

Authors 1, 1

Affiliations

  1. Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Abstract

We show that in the class of compact sets K in ℝn with an analytic parametrization of order m, the sets with Zariski dimension m are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on K.

Keywords

ENG
pluricomplex Green function, Siciak extremal function, traces of polynomials on semialgebraic sets, Zariski dimension, Bernstein and van der Corput-Schaake type inequalities

Bibliography

  1. [Ba1] M. Baran, Siciak's extremal function of convex sets in ℂn, Ann. Polon. Math. 48 (1988), 275-280.
  2. [Ba2] M. Baran, Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in ℝn, Michigan Math. J. 39 (1992), 395-404.
  3. [Ba3] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in ℝn, Proc. Amer. Math. Soc. 123 (1995), 485-494.
  4. [Ba4] M. Baran, Bernstein type theorems for compact sets in ℝn revisited, J. Approx. Theory 79 (1994), 190-198.
  5. [Ba5] M. Baran, Markov inequality on sets with polynomial parametrization, Ann. Polon. Math. 60 (1994), 69-79.
  6. [BaPl1] M. Baran and W. Pleśniak, Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves, Studia Math. 125 (1997), 83-96.
  7. [BaPl2] M. Baran and W. Pleśniak, Polynomial inequalities on algebraic sets, this issue, 209-219.
  8. [BeT] E. Bedford and B. A. Taylor, The complex equilibrium measure of a symmetric convex set in ℝn, Trans. Amer. Math. Soc. 294 (1986), 705-717.
  9. [BeRi] R. Benedetti and J.-J. Risler, Real Algebraic and Semi-Algebraic Sets, Hermann, Paris, 1990.
  10. [Bern] S. N. Bernstein, Sur l'ordre de la meilleure approximation des fonctions continues par des polynômes de degré donné, Mém. Acad. Roy. Belg. 4 (2) (1912), 1-103.
  11. [BLMT] L. Bos, N. Levenberg, P. Milman and B. A. Taylor, Tangential Markov inequalities characterize algebraic submanifolds of ℝN, Indiana Univ. Math. J. 44 (1995), 115-138.
  12. [BLT] L. Bos, N. Levenberg and B. A. Taylor, Characterization of smooth, compact algebraic curves in ℝ2, in: Topics in Complex Analysis, P. Jakóbczak and W. Pleśniak (eds.), Banach Center Publ. 31, Inst. Math. Polish Acad. Sci., Warszawa, 1995, 125-134.
  13. [BoMi1] L. Bos and P. Milman, On Markov and Sobolev type inequalities on compact subsets in ℝn, in: Topics in Polynomials in One and Several Variables and Their Applications, Th. Rassias et al. (eds.), World Scientific, Singapore, 1992, 81-100.
  14. [BoMi2] L. Bos and P. Milman, Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains, Geom. Funct. Anal. 5 (1995), 853-923.
  15. [Bru] A. Brudnyi, A Bernstein-type inequality for algebraic functions, Indiana Univ. Math. J. 46 (1997), 93-116.
  16. [CS1] J. G. van der Corput und G. Schaake, Ungleichungen für Polynome und trigonometrische Polynome, Compositio Math. 2 (1935), 321-361.
  17. [CS2] J. G. van der Corput und G. Schaake, Berichtigung zu: Ungleichungen für Polynome und trigonometrische Polynome, ibid. 3 (1936), 128.
  18. [DŁS] Z. Denkowska, S. Łojasiewicz et S. Stasica, Certaines propriétés élémentaires des ensembles sous-analytiques, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 529-536.
  19. [FeNa1] C. Fefferman and R. Narasimhan, Bernstein's inequality on algebraic curves, Ann. Inst. Fourier (Grenoble) 43 (1993), 1319-1348.
  20. [FeNa2] C. Fefferman and R. Narasimhan, On the polynomial-like behavior of certain algebraic functions, ibid. 44 (1994), 1091-1179.
  21. [FeNa3] C. Fefferman and R. Narasimhan, A local Bernstein inequality on real algebraic varieties, Math. Z. 223 (1996), 673-692.
  22. [Hir] H. Hironaka, Introduction to Real-Analytic Sets and Real-Analytic Maps, Istituto Matematico 'L. Tonelli', Pisa, 1973.
  23. [Jos] B. Josefson, On the equivalence between locally and globally polar sets for plurisubharmonic functions in ℂn, Ark. Mat. 16 (1978), 109-115.
  24. [K] M. Klimek, Pluripotential Theory, Oxford Univ. Press, London, 1991.
  25. [Lu] M. Lundin, The extremal plurisuhbarmonic function for the complement of convex subsets of ℝN, Michigan Math. J. 32 (1985), 196-201.
  26. [PaPl1] W. Pawłucki and W. Pleśniak, Markov's inequality and C∞ functions on sets with polynomial cusps, Math. Ann. 275 (1986), 467-480.
  27. [PaPl2] W. Pawłucki and W. Pleśniak, Extension of C∞ functions from sets with polynomial cusps, Studia Math. 88 (1988), 279-287.
  28. [Pl1] W. Pleśniak, Markov's inequality and the existence of an extension operator for C∞ functions, J. Approx. Theory 61 (1990), 106-117.
  29. [Pl2] W. Pleśniak, Recent progress in multivariate Markov inequality, in: Approximation Theory (In Memory of A. K. Varma), N. K. Govil et al. (eds.), Pure Appl. Math. 212, Marcel Dekker, New York, 1998, 449-464.
  30. [RoYo] N. Roytwarf and Y. Yomdin, Bernstein classes, Ann. Inst. Fourier (Grenoble) 47 (1997), 825-858.
  31. [Sa] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1983), 493-502.
  32. [Si1] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357.
  33. [Si2] J. Siciak, Extremal plurisubharmonic functions in ℂn, Ann. Polon. Math. 39 (1981), 175-211.
Studia Mathematica
2000/141/3
Export to PBN, Opens in new tab
Pages:
221-234
Main language of publication
English
Received
1998-11-23
Accepted
2000-04-03
Published
2000
Exact and natural sciences