Tame geometry with application in smooth analysis. (English) Zbl 1076.14079

Yomdin, Yosef; Comte, Georges

Lecture Notes in Mathematics 1834. Berlin: Springer (ISBN 3-540-20612-4/pbk). viii, 186 p. (2004).

This monograph is devoted to several related topics: geometry of real semialgebraic and tame sets, behavior of their “metric” characteristics under polynomial mappings, integral geometry, and geometry of critical and near critical values of differentiable mappings. Its subsequent chapters deal with the entropy dimension of a set, multidimensional variations and their use for bounding the \(\varepsilon\)-entropy of a set, the number of connected components of plane sections of semialgebraic sets, multidimensional variations of semialgebraic and tame sets, the multidimensional variations of images under polynomial mappings and a quantitative Sard theorem, quantitative transversality and cuspical values, quantitative Morse-Sard theorem bounding the \(\varepsilon\)-entropy of near-critical values, and finally some applications.
It is an interesting and useful contribution to the study of the quantitative aspects of the Morse-Sard theorem.

Reviewer: Jean Mawhin (Louvain-La-Neuve)

MSC:

14P10	Semialgebraic sets and related spaces
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14Q20	Effectivity, complexity and computational aspects of algebraic geometry
26B15	Integration of real functions of several variables: length, area, volume
32S15	Equisingularity (topological and analytic)

Keywords:

semialgebraic sets; tame sets; integral geometry