On degrees of non-roughness of real projective varieties
Following A. A. Andronov's ideas, D. A. Gudkov had introduced in 1965 the concept of degree of non-roughness in the circle of problems related to the Hilbert's 16th problem. This notion is derived from that of structural stability (following Andronov and Pontrjagin, Gudkov uses the term ''rough'' for the now usual ''stable'') and measures the degree of topological degeneration of an algebraic curve or a variety. For example, a real algebraic variety is of degree of non-roughness 0 iff every small perturbation of coefficients of its equations leads to an isotopic variety. This notion proved to be useful in the theory of plane real algebraic curves [see the survey by \textit{D. A. Gudkov}, Russ. Math. Surv. 29, No.4, 1-79 (1974; Zbl 0316.14018)] but the following question remained unanswered up to the paper under review: does each real plane algebraic curve have a well-defined and finite degree of non-roughness? The author answers this question and the similar one about higher dimensional varieties affirmatively. His main tool is the stratification theory of R. Thom. Indeed, he answers these questions in a more general setting of degree of non-roughness relatively to a semi-algebraic stratification of projective space. He also considers a complexified version of these questions.
scientific article; zbMATH DE number 3887841