1974/08/31 by D A Gudkov, D. A. Gudkov · 5 citations
Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Algebraic Geometry and Number Theory
paper · doi:10.1070/rm1974v029n04abeh001288
crossref issued 1974/08/31 · crossref published 1974/08/31 · crossref published-print 1974/08/31 · openalex publication_date 1974/08/31 · crossref created 2005/11/08 · crossref published-online 2007/10/16 · crossref deposited 2024/10/03 · openalex created_date 2025/10/10 · crossref indexed 2026/07/29 · openalex updated_date 2026/07/30
This article is a survey of the results on Hilbert's 16th problem from 1876 to the present. 1. HARNACK'S THEOREM. The number of branches of a non-singular curve of order m in RP2 does not exceed 1/2(m – 1)(m – 2) + 1. Harnack's method for constructing curves with the greatest number of branches (M-curves). The methods of Hilbert, Brusotti, and Wiman for constructing M-curves. M-curves on a quadric.2. The concepts of roughness and degrees of non-roughness. Brusotti's theorem on the independence of the simplifications (small corrections) of simple double points and cusps.3. The theorems of Petrovskii and Oleinik on algebraic curves and surfaces. The generalization due to Kharlamov.4. Sextic curves in RP2. Cubic and quartic surfaces in RP3.5. An oval of a non-singular curve is said to be odd if it lies inside an odd number of other ovals, and even otherwise. Let A be an M-curve of even order m, and suppose that the number of its even (odd) ovals is P(L). Then P – L (m/2)2mod 8. Broad generalizations were given by Rokhlin and others.6. Some conjectures.