Polynomial inequalities on algebraic sets

M. Baran; W. Pleśniak

ArticleOriginal scientific text

Title

Polynomial inequalities on algebraic sets

Authors ¹, ¹

Affiliations

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

Abstract

We give an estimate of Siciak's extremal function for compact subsets of algebraic varieties in

ℂ^{n}

(resp.

ℝ^{n}

). As an application we obtain Bernstein-Walsh and tangential Markov type inequalities for (the traces of) polynomials on algebraic sets.

Keywords

ENG

Bernstein-Walsh and tangential Markov type inequalities on algebraic sets, pluricomplex Green function, traces of polynomials on algebraic sets, Siciak's extremal function

[Ba] M. Baran, Markov inequality on sets with polynomial parametrization, Ann. Polon. Math. 60 (1994), 69-79.
[BaPl1] M. Baran and W. Pleśniak, Markov's exponent of compact sets in $ℂ^{n}$ , Proc. Amer. Math. Soc. 123 (1995), 2785-2791.
[BaPl2] M. Baran and W. Pleśniak, Bernstein and van der Corput-Schaake type inequalities on semialgebraic curves, Studia Math. 125 (1997), 83-96.
[BaPl3] M. Baran and W. Pleśniak, Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities, this issue, 221-234.
[BLMT1] L. Bos, N. Levenberg, P. Milman and B. A. Taylor, Tangential Markov inequalities characterize algebraic submanifolds of $ℝ^{N}$ , Indiana Univ. Math. J. 44 (1995), 115-138.
[BLMT2] L. Bos, N. Levenberg, P. Milman and B. A. Taylor, Tangential Markov inequalities on real algebraic varieties, ibid. 47 (1998), 1257-1271.
[BLT] L. Bos, N. Levenberg and B. A. Taylor, Characterization of smooth, compact algebraic curves in $ℝ^{2}$ , in: Topics in Complex Analysis, P. Jakóbczak and W. Pleśniak (eds.), Banach Center Publ. 31, Inst. Math., Polish Acad. Sci., Warszawa, 1995, 125-134.
[Bru] A. Brudnyi, A Bernstein-type inequality for algebraic functions, Indiana Univ. Math. J. 46 (1997), 93-116.
[FeNa1] C. Fefferman and R. Narasimhan, Bernstein's inequality on algebraic curves, Ann. Inst. Fourier (Grenoble) 43 (1993), 1319-1348.
[FeNa2] C. Fefferman and R. Narasimhan, On the polynomial-like behavior of certain algebraic functions, ibid. 44 (1994), 1091-1179.
[FeNa3] C. Fefferman and R. Narasimhan, A local Bernstein inequality on real algebraic varieties, Math. Z. 223 (1996), 673-692.
[Gen] L. Gendre, Inégalité de Markov tangentielle locale sur les courbes algébriques de $ℝ^{n}$ , Université Paul Sabatier de Toulouse, preprint, 1998.
[Iz] S. Izumi, A criterion for algebraicity on analytic set germs, Proc. Japan Acad. Ser. A 68 (1992), 307-309.
[Jos] B. Josefson, On the equivalence between locally and globally polar sets for plurisubharmonic functions in $ℂ^{n}$ , Ark. Mat. 16 (1978), 109-115.
[K] M. Klimek, Pluripotential Theory, Oxford Univ. Press, London, 1991.
[Ł] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, Basel, 1991.
[PaPl] W. Pawłucki and W. Pleśniak, Markov's inequality and $C^{\infty}$ functions on sets with polynomial cusps, Math. Ann. 275 (1986), 467-480.
[Pl1] W. Pleśniak, Invariance of the L-regularity of compact sets in $ℂ^{n}$ under holomorphic mappings, Trans. Amer. Math. Soc. 246 (1978), 373-383.
[Pl2] W. Pleśniak, Compact subsets of $ℂ^{n}$ preserving Markov's inequality, Mat. Vesnik 40 (1988), 295-300.
[Pl3] W. Pleśniak, Recent progress in multivariate Markov inequality, in: Approximation Theory, In Memory of A. K. Varma (N. K. Govil et al., eds.), Marcel Dekker, New York, 1998, 449-464.
[RoYo] N. Roytwarf and Y. Yomdin, Bernstein classes, Ann. Inst. Fourier (Grenoble) 47 (1997), 825-858.
[Sa] A. Sadullaev, An estimate for polynomials on analytic sets, Math. USSR-Izv. 20 (1983), 493-502.
[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.
[Si2] J. Siciak, Extremal plurisubharmonic functions in $ℂ^{n}$ , Ann. Polon. Math. 39 (1981), 175-211.
[Tou] J.-C. Tougeron, Idéaux de fonctions différentiables, Springer, Berlin, 1972.

Title

Polynomial inequalities on algebraic sets

Affiliations

Abstract

Keywords

Bibliography