Polygons of Petrović and Fine, algebraic ODEs, and contemporary mathematics (Q2212346)

The geometrically oriented contribution of Mihailo Petrović (1868--1943) to the theory of ordinary differential equations, in particular to generalizing the Newton-Puiseux polygonal method for algebraic ODEs, has been largely overlooked. Princeton mathematician \textit{H. B. Fine}'s cotemporaneous, independent work on the topic, ``On the functions defined by differential equations, with an extension of the Puiseux polygon construction to these equations'' [Am. J. Math. 11, 317--328 (1889; JFM 21.0302.01)] is cited in works such as \textit{V. V. Golubev}'s [Vorlesungen über die analytische Theorie der Differentialgleichungen (Russian). Moskau-Leningrad: Staatsverlag für technisch-theoretische Literatur (1941; Zbl 0061.16608)] where other parts of Petrović's 1894 thesis are noted. The authors conjecture that the topic may have failed to get more attention because Petrović's early students at the University of Belgrade who did research on this topic had careers interrupted by war and subsequent students did not pursue it. The 1894 thesis, written in French, was apparently not published in any other fashion until 1999, but his polygonal method appeared in 1899 [Acta Math. 22, 379--386 (1899; JFM 30.0293.01)]. The authors make some interesting comparisons between the lives of Fine and Petrović, noting that they both were key figures in raising the international stature in mathematics of their respective universities. Main attention is given to paraphrasing proofs from Petrović's thesis and from others who developed further aspects starting notably in the 1990s by such as J. Cano, P. Aroca, F. Jung and A. D. Bruno. Movable singularities of algebraic ODEs of the first order is a major application; Riccatti equations also play a role. The references provide an overview of the current state of the field. Though published in a history journal this largely mathematical exposition is explicitly intended not only for those who may wish to acknowledge the overlooked work of Petrović, but also for those wishing to gain new mathematical insights from his work.