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 largely 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, replied by means of a mix of the tools of actual semialgebraic and tame geometry and imperative geometry, grow to be nontrivial and hugely productive. The vital good thing about this process is that it permits the separation of the function of excessive differentiability and that of algebraic geometry in a gentle surroundings: all of the geometrically proper phenomena seem already for polynomial mappings. The geometric houses bought are "stable with admire to approximation", and will be imposed on soft features through polynomial approximation.
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