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

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Pith's one-line read A historical and mathematical study showing that Petrovic's and Fine's 1890s polygon methods generalize Newton-Puiseux theory and anticipate modern power geometry.

desk verdict A useful historical synthesis with one correct new observation, but the priority claim rests on a Briot–Bouquet citation left as a placeholder. read the letter →

arxiv 1908.03644 v2 pith:2GW36SOL submitted 2019-08-09 math.CA math.HO

Vladimir Dragovic , Irina Goryuchkina This is my paper · ORCID
classification math.CAmath.HO
keywords algebraicfinepetrovicequationsideasbeencontemporarydifferential
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Algebraic differential equations are equations built from polynomials in an unknown function and its derivatives. Around singular points, solutions can blow up to infinity, vanish, or branch like square roots. In the 1670s Newton invented a geometric 'polygon' trick for algebraic equations, later refined by Puiseux, that finds the first term of such expansions. This paper tells the story of two mathematicians, Henry Fine (1889) and Mihailo Petrovic (1894), who independently extended the polygon trick from algebraic equations to algebraic differential equations.

The idea is to plot each monomial of the equation as a point in a plane, with coordinates recording how many derivatives it contains and of what order. The convex hull of these points is a polygon, and each edge or vertex encodes a possible leading behavior of a solution. For first-order equations, Petrovic used this polygon to prove a clean criterion: the general solution has no movable zeros (or poles) exactly when the polygon has no slanted edge on the corresponding side. This is easier to check than the classical Fuchs conditions, which require solving a discriminant equation.

The paper also shows that Fine's polygon, which handles behavior at a singular point of the equation, is just Petrovic's polygon rotated by 90 degrees when the point is nonsingular. The authors connect these 1890s ideas to the contemporary 'power geometry' of Bruno and the work of Cano and Malgrange, arguing that much of the modern theory was anticipated a century earlier.

Extended reading notes

Core claim

The central assertion is that Petrovic's 1894 polygon method, independently paralleled by Fine in 1889, is a genuine generalization of the Newton-Puiseux method to algebraic ODEs and yields effective criteria for movable singularities. This is captured in Theorem 9: 'The necessary and sufficient condition for poles (zeros) of the general solution of a given algebraic ODE of the first order (25) not to depend on the constants of integration is that the polygon N of the equation (25) does not contain right (left) slanted edges.' If true, Petrovic established a simple geometric test decades before modern power geometry.

Load-bearing premise

The load-bearing historical premise is that the authors' reading of the primary sources ([67] Petrovic 1894, [29] Fine 1889, [70] collected works) is faithful and that Fine and Petrovic developed their methods independently and without knowledge of each other. This premise enters in Sections 1.1 and 1.2 and underpins the paper's novelty claims. If, for example, earlier work by Briot and Bouquet already contained the polygon construction for ODEs (hinted at in the incomplete citation 'see [?]' in the retrospective), the priority and neglect narrative would need revision.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript is a historical survey of geometric methods in the analytic theory of algebraic ordinary differential equations, centered on Mihailo Petrovic's 1894 polygon method and Henry Fine's 1889 independent construction. It presents the Petrovic polygon as a generalization of the Newton-Puiseux method from algebraic equations to algebraic ODEs, reconstructs both constructions with explicit coordinates, and proves a relation between them: Theorem 7 states that, for a nonsingular point, the Fine polygon of the translated equation coincides with the Petrovic polygon rotated by a quarter turn. The later sections survey applications: Petrovic's criteria for absence of movable poles and zeros (Theorem 9), his theorem on rational first-order ODEs with at most three essentially distinct transcendental single-valued solutions (Theorem 12), results on binomial ODEs, and the relation to modern power geometry, Painleve equations, and works of Cano, Bruno, and others. The final 'Retrospective' asserts that the first Newton-Puiseux-type ideas for differential equations may go back to Briot-Bouquet, but the relevant citation is left incomplete.

Significance. If the historical narrative is correct, the paper recovers a genuinely forgotten anticipation of modern power geometry: Petrovic's polygon test for movable poles and zeros is simpler to apply than Fuchs's discriminant-based criterion and predates the modern 'power geometry' literature by nearly a century. The paper's mathematical core is transparent and checkable: the coordinate definitions in Sections 3 and 4 are explicit, Theorem 7 follows directly from them, and Examples 1-3 contain enough detail for the reader to verify the leading asymptotics. The authors are also commendably candid about the limitations of planar polygons for higher-order equations (Section 3.1) and about Petrovic's absence from the Painleve program (Section 10.1). The main weakness is historical, not mathematical: the priority and neglect claims currently rest on incomplete citations, so the contribution is valuable but not yet in final form.

major comments (3)
  1. [Retrospective (unnumbered, before Acknowledgements)] The retrospective contains two unresolved citation placeholders: 'Broit and Bouquet, see [?]' and 'Cano ... see [ ?]'. The first is load-bearing: the abstract's claim that Petrovic's geometric ideas were 'left completely unnoticed by the experts' and the attribution of the ODE polygon method to Fine and Petrovic presuppose that no earlier work, in particular Briot-Bouquet, already contained a Newton-Puiseux polygon construction for differential equations. The authors must locate the relevant Briot-Bouquet passage, state explicitly whether it contains such a construction, and adjust the priority and neglect claims accordingly. The incomplete Cano citation should also be completed.
  2. [Sections 1.1-1.2] The independence of Fine and Petrovic, stated as 'independently' in the abstract and elaborated in Section 1.2, is not supported by documentary evidence; the text only notes that neither cited the other. Because independence is part of the paper's novelty narrative, the authors should either provide evidence such as correspondence, library records, or contemporary reviews, or explicitly qualify the claim as an inference from the absence of mutual citation.
  3. [Section 6, Theorem 9] Theorem 9, the paper's central mathematical exhibit, is presented with only a two-sentence proof sketch: sufficiency is deferred to Proposition 1 and necessity to 'the methods of analytic theory,' with a reference to Fuchs's techniques, without spelling out the argument. Since the paper states it as a theorem in its own text and uses it to claim that Petrovic anticipated modern criteria, please provide a complete proof or an exact page reference to [67] where Petrovic's proof is given, and explain how Painleve's Theorem 6 enters. This is needed for the reader to verify the 'necessary' direction.
minor comments (4)
  1. [Retrospective] The name 'Broit and Bouquet' is a typo for 'Briot and Bouquet'; elsewhere the spelling is correct.
  2. [References, [25]] Reference [25] is described as 'to appear, Bulletin of the AMS, 2020'; if the paper has appeared by now, the reference should be updated with full publication data.
  3. [Section 3 and Theorem 10] The term 'angular coefficient' is used repeatedly without a definition; define it explicitly as the slope of an edge in the (M,N)-plane and state its relation to the exponent lambda in (15).
  4. [Section 6, paragraph before Theorem 9] The sentence 'each its vertex corresponds to the exactly one monomial' is grammatically garbled and should be rewritten; a one-sentence justification for first-order equations would also help, since for different terms with the same (M,N) the correspondence is to a unique monomial only after summing coefficients.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a historical survey whose only new coordinate comparison is self-contained, and its self-citations are illustrative rather than load-bearing.

full rationale

No circular step meets the evidence standard. The paper's central mathematical content is historical attribution: Theorem 9 is presented as Petrovic's 1894 criterion, with sufficiency following from Proposition 1 and necessity from Fuchs-type arguments, not from any parameter fitted to the conclusion. The only new derivation, Theorem 7, is an explicit coordinate comparison: substituting x = z + x0 sends Petrovic's exponent pair (M_i,N_i) to Fine's pair (-N_i,M_i), so the claimed coincidence is a direct computation, not an assumed equivalence. The authors' own works [9], [10], [11], and [25] appear only as examples of later Power Geometry applications and as a BAMS cover note; they do not support the priority claim or the polygon criteria. The retrospective does contain unresolved historical citations: 'The first ideas to use the Newton – Puiseux methods in the theory of differential equations probably goes back to Broit and Bouquet, see [?]' and 'Cano implemented the ideas ... see [?].' These are missing references and leave the Briot-Bouquet precedence question open, but an incomplete citation is an evidentiary/completeness concern, not a circular reduction: the mathematical claims do not define their conclusion in terms of that citation. No equation is defined in terms of the result it is used to prove, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or new entities. It relies on classical theorems of analytic ODE theory and on an unverified historical premise about the contents and reception of 1890s publications.

assumptions (4)
  • standard math Painleve Theorem 6: first-order algebraic ODEs have no movable non-algebraic singularities.
    Invoked in Section 3.1 and Section 6 to justify that planar polygons capture all movable singularities for first-order equations.
  • standard math Fuchs Theorem 5 (necessary and sufficient conditions for absence of movable critical points).
    Used as background and in the proof sketch of Petrovic's Theorem 9.
  • domain assumption Historical accuracy of primary sources [67], [29], [30], [70] as read by the authors.
    The central historical narrative (Petrovic's neglect, Fine's independent discovery, no mutual awareness) rests on the authors' reading of these sources and on the absence of contrary citations.
  • standard math Hermite Theorem 3 and Briot-Bouquet theory for elliptic solutions.
    Used in Sections 7 and 9 for classifying single-valued and fixed-singularity solutions.

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Pith. "Pith review of Polygons of Petrovic and Fine, algebraic ODEs, and contemporary mathematics." pith.science (2026). https://pith.science/paper/2GW36SOL

@misc{pith2026190803644,
  author       = {Pith},
  title        = {Pith review of: Polygons of Petrovic and Fine, algebraic ODEs, and contemporary mathematics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2GW36SOL}},
  note         = {Machine review of arXiv:1908.03644}
}
read the original abstract

Here, 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 Petrovic on algebraic differential equations and in particular his geometric ideas captured in his polygon method from the last years of the XIXth century, which have been left completely unnoticed by the experts. This concept, also developed in a bit a different direction and independently by Henry Fine, generalizes the famous Newton-Puiseux polygonal method and applies to algebraic ODEs rather than algebraic equations. Although remarkable, the Petrovic legacy has been practically neglected in the modern literature, while the situation is less severe in the case of results of Fine. Thus, we study the development of the ideas of Petrovic and Fine and their places in contemporary mathematics.

Figures

Figures reproduced from arXiv: 1908.03644 by the authors.

Figure 1
Figure 1. Construction of Petrovi´c polygon on vertices. The upper half-plane of the plane MON can be decomposed on rays with the angular coefficients λ and the open angular sectors, containing rays with angular coefficients λ, where λ are all the values between the angular coefficients of the edges meeting at the given vertex, see [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. The Petrovi´c polygons of: a) the equation (22); b) the equation (24). [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. The Petrovi´c polygon of the equation (26). [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The Petrovi´c polygon of the equation (28). [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 5
Figure 5. Figure 5: The Petrovi´c polygon of the equation (43): a) [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]

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