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 not often turns into obvious. the most cause is that the classical Morse-Sard theorem is essentially qualitative. This quantity supplies a proof and additionally an "explanation" of the quantitative Morse-Sard theorem and similar effects, starting with the study of polynomial (or tame) mappings. The quantitative questions, replied by way of a mixture of the tools of actual semialgebraic and tame geometry and essential geometry, turn into nontrivial and hugely productive. The vital benefit of this method is that it permits the separation of the function of excessive differentiability and that of algebraic geometry in a delicate environment: all of the geometrically suitable phenomena look already for polynomial mappings. The geometric houses got are "stable with appreciate to approximation", and will be imposed on soft features through polynomial approximation.
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