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Polygons of Petrović and Fine, algebraic ODEs, and contemporary mathematics

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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.

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Notes

  1. Let us briefly mention basic biographic data about Lazarus Fuchs. He was born in 1833 in Moschina, the Grand Duchy of Posen of Kingdom of Prussia, nowadays Poland. He worked on his PhD in Berlin University with Kummer as his advisor, from 1854 till 1858, when he defended a thesis on the lines of curvature on surfaces. His interest in differential equations came from his association with Weierstrass (Gray 1984). In 1882 he returned to Berlin where he got a position of a full professor of the Berlin University. He was elected a member of Berlin Academy in 1884. From 1892 till his death, Fuchs served as the Editor-in-Chief of “Journal für die reine und angewandte Mathematik” (Crelle’s journal). He died in Berlin in 1902.

  2. Paul Painlevé was born in Paris in 1863. He graduated from the École Normale in 1877 and went on to become a full professor of the École Normale and Sorbonne. He was an elected member of the French Academy since 1900. After 1910, and election to the national parliament, Painlevé shifted his focus from science to politics. He was a minister of several French governments, including the post of the Minster of War during the World War I. Painlevé served as the Prime Minister of France two times: September 12–November 16, 1917, and April 17–November 28, 1925. Painlevé died in Paris in 1933.

  3. Kovalevskaya’s name on her published papers is given as Kowalevski but she is often called Kovalevskaya in the current literature, following English transcription of her Russian family name.

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Acknowledgements

The authors are grateful to Renat Gontsov for numerous valuable comments and suggestions. They are also grateful to Joseph Minich and Borislav Gajić for careful reading of the manuscript and their suggestions. The authors would like to thank Professor Susan Friedlander and the Bulletin of the AMS for kindly providing them the opportunity to feature this paper at the cover page of one of the forthcoming issues of BAMS [see Dragović and Goryuchkina (2020)]. The first author would like to thank Professors Božidar Jovanović, Zoran Petrić, and Vera Kovačević-Vujičić, the heads of the departments for Mechanics, for Mathematics, and for Applied Mathematics and Computer Science for the invitation to deliver a talk about the work of Mihailo Petrović at the first joint meeting of all three colloquia of the Mathematical Institute of the Serbian Academy of Sciences and Arts and to Academicians Stevan Pilipović and Gradimir Milovanović, the co-chairs of the public manifestation in Serbia “2018-the Year of Mihailo Petrović Alas” for the invitation to talk at the round table on May 22, 2018, at the Serbian Academy of Sciences and Arts, see Dragović (2019).

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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080, USA

    Vladimir Dragović

  2. Mathematical Institute SANU, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Vladimir Dragović

  3. Keldysh Institute of Applied Mathematics of RAS, Moscow, Russia

    Irina Goryuchkina

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  1. Vladimir Dragović
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Correspondence to Vladimir Dragović.

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The research of the first author has been partially supported by the Grant 174020 “Geometry and topology of manifolds, classical mechanics, and integrable dynamical systems” of the Ministry of Education and Sciences of Serbia and by the University of Texas at Dallas. The second author gratefully acknowledges the Grant PRAS-18-01 (PRAN 01 “Fundamental mathematics and its applications”).

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Dragović, V., Goryuchkina, I. Polygons of Petrović and Fine, algebraic ODEs, and contemporary mathematics. Arch. Hist. Exact Sci. 74, 523–564 (2020). https://doi.org/10.1007/s00407-020-00250-3

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