Authors: Dragović, Vladimir 
Goryuchkina, Irina
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Polygons of Petrović and Fine, algebraic ODEs, and contemporary mathematics
Journal: Archive for History of Exact Sciences
Issue Date: 1-Jan-2020
Rank: M22
ISSN: 0003-9519
DOI: 10.1007/s00407-020-00250-3
Abstract: 
In this paper, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary differential equations. This leads us to the work of Mihailo Petrović on algebraic differential equations (ODEs) and in particular the geometric ideas expressed in his polygon method from the final years of the nineteenth century, which have been left completely unnoticed by the experts. This concept, also developed independently and in a somewhat different direction by Henry Fine, generalizes the famous Newton–Puiseux polygonal method and applies to algebraic ODEs rather than algebraic equations. Although remarkable, the Petrović legacy has been practically neglected in the modern literature, although the situation is less severe in the case of results of Fine. Therefore, we study the development of the ideas of Petrović and Fine and their places in contemporary mathematics.
Publisher: Springer Link
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Grant PRAS-18-01 (PRAN 01 “Fundamental mathematics and its applications”)

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