Abstract
The paper contains a full geometric characterization of compact semialgebraic sets in C satisfying the Łojasiewicz-Siciak condition. The Łojasiewicz-Siciak condition is a certain estimate for the Siciak extremal function. In a previous paper, we gave a sufficient criterion for a compact, connected, and semialgebraic set in C to satisfy this condition. In the present paper, we remove completely the connectedness assumption and prove that the aforementioned sufficient condition is also necessary. Moreover, we obtain some new results concerning the Łojasiewicz-Siciak condition in CN. For example, we prove that if K 1,...,K p are compact, nonpluripolar, and pairwise disjoint subsets of CN, each satisfying the Łojasiewicz-Siciak condition, and K:= K 1⋃· · ·⋃K p is polynomially convex, then K satisfies this condition as well.
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References
E. Bedford and B. A. Taylor, Plurisubharmonic functions with logarithmic singularities, Ann. Inst. Fourier (Grenoble), 38 (1988), 133–171.
L. Bialas-Ciez and M. Kosek, Iterated function systems and Lojasiewicz-Siciak condition of Green’s function, Potential Anal., 34 (2011), 207–221.
E. Bierstone and P. Milman, Semianalytic and subanalytic sets, Publ. Math. Inst. Hautes études Sci., 67 (1988), 5–42.
J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Springer-Verlag, Berlin, 1998.
M. Coltoiu, Complete locally pluripolar sets, J. Reine Angew. Math., 412 (1990), 108–112.
L. van den Dries, Tame Topology and o-minimal Structures, Cambridge University Press, Cambridge, 1998.
L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J., 84 (1996), 497–540.
F. Forstnerić, Stein Manifolds and Holomorphic Mappings, Springer, Heidelberg, 2011.
L. Gendre, Inégalités de Markov singulières et approximation des fonctions holomorphes de la classe M, Ph. D. thesis, Université Toulouse III (Paul Sabatier), 2005.
R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, John Wiley & Sons, New York, 1997.
L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland Publishing Co., Amsterdam, 1990.
T. Kaiser, Capacity density of subanalytic sets in higher dimension, Potential Anal., 20 (2007), 397–407.
T. Kaiser, Dirichlet regularity of subanalytic domains, Trans. Amer. Math. Soc., 360 (2008), 6573–6594.
T. Kaiser, The Dirichlet problem in the plane with semianalytic raw data, quasianalyticity and o-minimal structures, Duke Math. J., 147 (2009), 285–314.
E. Kallin, Polynomial convexity: The three spheres problem, Proc. Conf. Complex Analysis, Springer-Verlag, Berlin, 1965, pp. 301–304.
M. Klimek, Pluripotential Theory, Oxford University Press, New York, 1991.
F. Leja, Teoria funkcji analitycznych, PWN, Warsaw, 1957.
N. Levenberg, Approximation in CN, Surv. in Approx. Theory, 2 (2006), 92–140.
S. Lojasiewicz, Ensembles semi-analytiques, IHES, Bures-sur-Yvette, 1965.
S. Lojasiewicz, An Introduction to the Theory of Real Functions, 3rd ed., John Wiley & Sons, Chichester, 1988.
D. Ma and T. Neelon, On convergence sets of formal power series, Complex Analysis and its Synergies, 1 (2015), 1–21.
C. Miller, Exponentiation is hard to avoid, Proc. Amer. Math. Soc., 122 (1994), 257–259.
R. Näkki and B. Palka, Boundary angles, cusps and conformal mappings, Complex Var. Theory, Appl., 5 (1986), 165–180.
W. Pawlucki and W. Plésniak, Markov’s inequality and C8 functions on sets with polynomial cusps, Math. Ann., 275 (1986), 467–480.
R. Pierzchala, UPC condition in polynomially bounded o-minimal structures, J. Approx. Theory 132 (2005), 25–33.
R. Pierzchala, Markov’s inequality in the o-minimal structure of convergent generalized power series, Adv. Geom., 12 (2012), 647–664.
R. Pierzchala, The Lojasiewicz-Siciak condition of the pluricomplex Green function, Potential Anal., 40 (2014), 41–56.
R. Pierzchala, An estimate for the Siciak extremal function–Subanalytic geometry approach, J. Math. Anal. Appl., 430 (2015), 755–776.
W. Plésniak, A criterion for polynomial conditions of Leja’s type in CN, Univ. Iagel. Acta Math. 24 (1984), 139–142.
W. Plésniak, L-regularity of subanalytic sets in Rn, Bull. Polish Acad. Sci. Math. 32 (1984), 647–651.
W. Plésniak, Pluriregularity in polynomially bounded o-minimal structures, Univ. Iagel. Acta Math. 41 (2003), 205–214.
W. Plésniak, Siciak’s extremal function in complex and real analysis, Ann. Polon. Math., 80 (2003), 37–46.
[33] Ch. Pommerenke, Conformal maps at the boundary, Hand book of Complex Analysis: Geometric Function Theory (vol. 1), Elsevier, 2002, 37–74.
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.
J. Siciak, Extremal plurisubharmonic functions in Cn, Ann. Polon. Math., 39 (1981), 175–211.
J. Siciak, Rapid polynomial approximation on compact sets in CN, Univ. Iagel. Acta Math., 30 (1993), 145–154.
E. L. Stout, Polynomial Convexity, Birkhäuser Boston, Boston, MA, 2007.
V. P. Zaharjuta, Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernstein-Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math., 33 (1976), 137–148.
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Pierzchała, R. Semialgebraic sets and the Łojasiewicz-Siciak condition. JAMA 129, 285–307 (2016). https://doi.org/10.1007/s11854-016-0022-z
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DOI: https://doi.org/10.1007/s11854-016-0022-z