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

Baran, Mirosław; Pleśniak, W.

doi:10.4064/sm-141-3-221-234

Cited by 10 publications

(12 citation statements)

References 19 publications

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“…In [8,9] and in [14,[3][4][5] this problem has been connected with some results in potential theory and with a characterization of algebraic subsets. In [50,12,27,28,59] Bernstein classes have been used in counting zeroes in finite dimensional families of analytic functions (this problem is closely related to the classical problem of counting closed trajectories, or "limit cycles" of plane polynomial vector fields).…”

Section: Bernstein-type Inequalities For Algebraic Functionsmentioning

confidence: 98%

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Analytic reparametrization of semi-algebraic sets

Yomdin

2008

Journal of Complexity

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In many problems in analysis, dynamics, and in their applications, it is important to subdivide objects under consideration into simple pieces, keeping control of high-order derivatives. It is known that semi-algebraic sets and mappings allow for such a controlled subdivision: this is the "C k reparametrization theorem" which is a high-order quantitative version of the well-known results on the existence of a triangulation of semialgebraic sets. In a C k -version we just require in addition that each simplex be represented as an image, under the "reparametrization mapping" , of the standard simplex, with all the derivatives of up to order k uniformly bounded. The main result of this paper is, that if we reparametrize all the set A but its small part of a size , we can do much more: not only to "kill" the derivatives, but also to bound uniformly the analytic complexity of the pieces, while their number remains of order log 1 . SummaryQuantitative information about geometric and analytic structure of algebraic and semi-algebraic sets is important in many problems of analysis, geometry, differential equations, dynamics, etc. In some applications it is enough to control just the rough topological information, like the number of simplices in the triangulation. In others the Lipschitzian or the C 1 bounds are important (see, for example, [40,41]). In some problems of analysis and dynamics the control of highorder derivatives is essential. It turns out that semi-algebraic sets and mappings allow for such a control.The main example is provided by the "C k reparametrization theorem" [55][56][57]34]. This can be considered as a high-order quantitative version of the well known result on the existence of a ଁ

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Section: Bernstein-type Inequalities For Algebraic Functionsmentioning

confidence: 98%

“…Bernstein-type inequalities on algebraic sets have been intensively investigated in the last decade (see [3][4][5]8,9,12,14,[23][24][25]27,28,50,59]). …”

Section: Analytic Reparametrizationmentioning

confidence: 99%

Analytic reparametrization of semi-algebraic sets

Yomdin

2008

Journal of Complexity

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“…It also plays a fundamental role in characterizing compact pieces of an algebraic variety in C n in terms of tangential Markov, Bernstein or van der Corput-Schaake inequalities , Bos, Levenberg, Milman and Taylor 1995, Baran and Pleśniak 1997, 2000b. Let us add that the techniques developed in Baran and Pleśniak (2000b) are based on fine bounds for Siciak's extremal func-tion associated with a ball in R n which are due to Baran (1988Baran ( , 1992Baran ( , 1998 and which have been inspired by Lundin (1985), who solved an old question asked by Siciak about a formula for the extremal function associated with the Euclidean unit ball in R n .…”

Section: and If Int E Is Dense In E Then E Is (Locally) L-regular mentioning

confidence: 99%

Siciak's extremal function in complex and real analysis

Pleśniak

2003

Ann. Polon. Math.

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Abstract. The Siciak extremal function establishes an important link between polynomial approximation in several variables and pluripotential theory. This yields its numerous applications in complex and real analysis. Some of them can be found on a rich list drawn up by Klimek in his well-known monograph "Pluripotential Theory". The purpose of this paper is to supplement it by applications in constructive function theory.The extremal function associated with a compact subset E of C n is defined by the formulaIt was introduced more than forty years ago by Józef Siciak who used it to prove a multivariate version of the Bernstein-Walsh theorem on polynomial approximation of germs of holomorphic functions on compact subsets of C n (Siciak 1962). It is known (Zakharyuta 1976a, Siciak 1981 wherewhereas |z| → ∞} is the Lelong class of plurisubharmonic functions with logarithmic growth at infinity. If E is nonpluripolar, then the plurisubharmonic functionis the unique function in the class L(C n ) which vanishes on E except possibly on a pluripolar subset and satisfies the complex Monge-Ampère equation

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“…Important applications in Computational Geometry have been proposed ( [34,72,73,74] and Section 5.3 below). Also in Approximation Theory (especially in study of Polynomial Inequalities on algebraic sets) importance of parametrizations was well recognized ( [2,3,4,83] and Section 5.4 below).…”

Section: Introductionmentioning

confidence: 99%

Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

Yomdin

2015

Japan J. Indust. Appl. Math.

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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.

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

Cited by 10 publications

References 19 publications

Analytic reparametrization of semi-algebraic sets

Analytic reparametrization of semi-algebraic sets

Siciak's extremal function in complex and real analysis

Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

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