By Yosef Yomdin
The Morse-Sard theorem is a slightly subtle result and the interaction among the high-order analytic constitution of the mappings concerned and their geometry hardly turns into obvious. the most cause is that the classical Morse-Sard theorem is essentially qualitative. This quantity provides a proof and additionally an "explanation" of the quantitative Morse-Sard theorem and comparable effects, starting with the study of polynomial (or tame) mappings. The quantitative questions, spoke back via a mixture of the equipment of actual semialgebraic and tame geometry and essential geometry, turn into nontrivial and hugely productive. The very important good thing about this process is that it permits the separation of the function of excessive differentiability and that of algebraic geometry in a tender surroundings: the entire geometrically correct phenomena look already for polynomial mappings. The geometric houses bought are "stable with recognize to approximation", and will be imposed on delicate features through polynomial approximation.
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