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 similar effects, starting with the study of polynomial (or tame) mappings. The quantitative questions, replied by means of a mixture of the equipment of actual semialgebraic and tame geometry and fundamental geometry, turn into nontrivial and hugely productive. The vital good thing about this process is that it permits the separation of the position of excessive differentiability and that of algebraic geometry in a soft atmosphere: the entire geometrically correct phenomena look already for polynomial mappings. The geometric houses acquired are "stable with admire to approximation", and will be imposed on tender features through polynomial approximation.