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

M. Baran; W. Pleśniak

Studia Mathematica (2000)

  • Volume: 141, Issue: 3, page 221-234
  • ISSN: 0039-3223

Abstract

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We show that in the class of compact sets K in n 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.

How to cite

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Baran, M., and Pleśniak, W.. "Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities." Studia Mathematica 141.3 (2000): 221-234. <http://eudml.org/doc/216781>.

@article{Baran2000,
abstract = {We show that in the class of compact sets K in $ℝ^n$ 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.},
author = {Baran, M., Pleśniak, W.},
journal = {Studia Mathematica},
keywords = {pluricomplex Green function; Siciak extremal function; traces of polynomials on semialgebraic sets; Zariski dimension; Bernstein and van der Corput-Schaake type inequalities},
language = {eng},
number = {3},
pages = {221-234},
title = {Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities},
url = {http://eudml.org/doc/216781},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Baran, M.
AU - Pleśniak, W.
TI - Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 3
SP - 221
EP - 234
AB - We show that in the class of compact sets K in $ℝ^n$ 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.
LA - eng
KW - pluricomplex Green function; Siciak extremal function; traces of polynomials on semialgebraic sets; Zariski dimension; Bernstein and van der Corput-Schaake type inequalities
UR - http://eudml.org/doc/216781
ER -

References

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  1. [Ba1] M. Baran, Siciak’s extremal function of convex sets in n , Ann. Polon. Math. 48 (1988), 275-280. Zbl0661.32023
  2. [Ba2] M. Baran, Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in n , Michigan Math. J. 39 (1992), 395-404. Zbl0783.32009
  3. [Ba3] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in n , Proc. Amer. Math. Soc. 123 (1995), 485-494. Zbl0813.32011
  4. [Ba4] M. Baran, Bernstein type theorems for compact sets in n revisited, J. Approx. Theory 79 (1994), 190-198. Zbl0819.41013
  5. [Ba5] M. Baran, Markov inequality on sets with polynomial parametrization, Ann. Polon. Math. 60 (1994), 69-79. Zbl0824.41014
  6. [BaPl1] M. Baran and W. Pleśniak, Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves, Studia Math. 125 (1997), 83-96. Zbl0895.41011
  7. [BaPl2] M. Baran and W. Pleśniak, Polynomial inequalities on algebraic sets, this issue, 209-219. Zbl0987.41006
  8. [BeT] E. Bedford and B. A. Taylor, The complex equilibrium measure of a symmetric convex set in n , Trans. Amer. Math. Soc. 294 (1986), 705-717. Zbl0595.32022
  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 N , Indiana Univ. Math. J. 44 (1995), 115-138. Zbl0824.41015
  12. [BLT] L. Bos, N. Levenberg and B. A. Taylor, Characterization of smooth, compact algebraic curves in 2 , in: Topics in Complex Analysis, P. Jakóbczak and W. Pleśniak (eds.), Banach Center Publ. 31, Inst. Math. Polish Acad. Sci., Warszawa, 1995, 125-134. Zbl0834.41012
  13. [BoMi1] L. Bos and P. Milman, On Markov and Sobolev type inequalities on compact subsets in n , 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. Zbl0848.46022
  15. [Bru] A. Brudnyi, A Bernstein-type inequality for algebraic functions, Indiana Univ. Math. J. 46 (1997), 93-116. Zbl0876.26015
  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. Zbl0013.10803
  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. Zbl0435.32006
  19. [FeNa1] C. Fefferman and R. Narasimhan, Bernstein's inequality on algebraic curves, Ann. Inst. Fourier (Grenoble) 43 (1993), 1319-1348. Zbl0842.26013
  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. Zbl0911.32011
  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 n , Ark. Mat. 16 (1978), 109-115. Zbl0383.31003
  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 N , 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. Zbl0579.32020
  27. [PaPl2] W. Pawłucki and W. Pleśniak, Extension of C functions from sets with polynomial cusps, Studia Math. 88 (1988), 279-287. Zbl0778.26010
  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. Zbl0984.41007
  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. Zbl0582.32023
  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. Zbl0111.08102
  33. [Si2] J. Siciak, Extremal plurisubharmonic functions in n , Ann. Polon. Math. 39 (1981), 175-211. Zbl0477.32018

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