On degrees of non-roughness of real projective varieties
type: Item
instance of: scholarly article
title: On degrees of non-roughness of real projective varieties
zbMATH Open document ID: 0557.14012
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 32C05
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 3887841
zbMATH Keywords: real projective varieties
zbMATH Keywords: degree of non-roughness
zbMATH Keywords: stability
zbMATH Keywords: semi- algebraic stratification
author: Goo Ishikawa
MaRDI profile type: Publication
full work available at URL: pjaa.59.277
OpenAlex ID: W1967801651
cites work: Topological stability of smooth mappings
cites work: THE TOPOLOGY OF REAL PROJECTIVE ALGEBRAIC VARIETIES
cites work: Q4773093
cites work: Q3037597
cites work: COMPLEX TOPOLOGICAL CHARACTERISTICS OF REAL ALGEBRAIC CURVES
cites work: Ensembles et morphismes stratifiés
DOI: 10.3792/PJAA.59.277
description: scientific article; zbMATH DE number 3887841