Pith. sign in

REVIEW 2 major objections 4 minor 18 references

Existence and convergence of Puiseux series solutions for autonomous first order differential equations

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that every formal Puiseux series solution of an autonomous first-order algebraic ODE converges, and gives an algorithm enumerating all such solutions.

desk verdict Genuinely extends convergence of formal solutions to Puiseux series, but the infinity case is not proved as written because of the sign change; fixable, deserves review. read the letter →

arxiv 1908.09196 v2 pith:BBLKRZRJ submitted 2019-08-24 math.AG

Jose Cano , Sebastian Falkensteiner , J. Rafael Sendra This is my paper · ORCID
classification math.AG MSC 34A0934M2514H20
keywords algebraicdifferentialequationcurveplaceformalPuiseuxseriessolutionconvergentautonomousfirstorderODENewtonpolygonmethodBriot-Bouquetlemma
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

This paper proves that for an algebraic ordinary differential equation F(y,y')=0, any formal solution expressed as a Puiseux series, with fractional powers of the independent variable and expanded around any finite point or at infinity, actually converges to an analytic function. The result extends the previously known case of ordinary power series solutions to fractional-power solutions by connecting each formal solution to a place of the algebraic curve F(y,p)=0. The connection is constructive: the authors give an algorithm, built on place computations and an associated first-order equation, that lists all Puiseux solutions. A direct consequence is that for every point (x0,y0) in the complex plane there is an analytic solution curve passing through that point. If the proof is right, formal fractional-power solutions of these equations never need to be treated as divergent formal objects.

What carries the argument

The load-bearing object is the solution-place correspondence: map a formal Puiseux solution y(x) of ramification order n to the irreducible parametrization (a(t), b(t)) = (y(t^n), $t^{{hn}}$ y'(t^n)) of the curve F(y,p)=0, whose equivalence class is a place. The carrying identity is a'(t) = n $t^{{n(1-h)-1}}$ b(t), equivalent to requiring that the reparametrization solves the associated first-order equation (3.5); that equation is transformed by a change of variable into the Briot-Bouquet form g(t,z) t z' = f(t,z), a classical existence-and-convergence result for such equations. The Briot-Bouquet lemma supplies uniqueness and parameter counts as well as the convergence conclusion, and Lemma 3.9 transfers those properties back to the reparametrization and hence to the Puiseux solution.

What would settle it

Find an autonomous first-order algebraic equation whose associated reparametrization equation at infinity, in the resonant case, has a formal power series solution whose coefficient sequence grows faster than any geometric series; such a sequence would be visible in the recurrence generated by Lemma 3.9 and would contradict the claimed convergence of all formal Puiseux solutions.

Watch

Extended reading notes

Core claim

On the paper's own terms: every formal Puiseux series solution of an autonomous first-order algebraic differential equation, expanded at a finite point or at infinity, is convergent. The proof runs through the associated algebraic curve: a solution y(x) of ramification order n determines an irreducible formal parametrization (a(t), b(t)) = (y(t^n), $t^{{hn}}$ y'(t^n)), and a place of the curve is a solution place exactly when the reparametrization s(t) satisfies the associated first-order equation a'(s(t)) s'(t) = n $t^{{n(1-h)-1}}$ b(s(t)). Applying the Briot-Bouquet lemma to this associated equation shows the reparametrization is convergent, hence the original Puiseux series is convergent. The paper also claims a converse for expansions at finite points, where the order condition is necessary and sufficient, and gives an algorithmic description of all solutions; at infinity the order condition is necessary but not sufficient, and the algorithm checks solvability directly.

Load-bearing premise

The proof depends on the resonant case of the classical Briot-Bouquet lemma, where the eigenvalue is a positive integer and convergence is obtained by a change of variables that the paper cites rather than proves; if that reduction were invalid, the convergence result at infinity would be unsupported.

Editorial extensions

If this is right

  • At finite points, every formal power or fractional-power solution of F(y,y')=0 is in fact an analytic solution, so no divergent formal branch of this type exists.
  • For any point (x0,y0) in the complex plane, there is a local analytic solution curve of F(y,y')=0 passing through it.
  • All Puiseux solutions expanded around zero can be listed algorithmically, and the computed truncations stand in one-to-one correspondence with the true solutions.
  • At infinity, every computed solution truncation extends to a genuine solution, but uniqueness is not guaranteed; one-parameter families of solutions at infinity can occur.
  • The number of solution parametrizations through a point is bounded in terms of the degree of F in p, which rules out certain infinite families such as y = x + c x^2 as solutions of any such equation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the convergence theorem is correct, a natural test is whether non-autonomous first-order equations or higher-order autonomous equations admit genuinely divergent Puiseux solutions, since the place-based proof does not directly apply there.
  • The majorant-series estimates inside the Briot-Bouquet lemma could likely be sharpened to give explicit lower bounds on the radius of convergence of each constructed solution, a quantitative consequence the paper does not state.
  • The free parameter appearing in the infinity case likely corresponds to true analytic families of solutions at infinity; checking whether distinct parameter values always produce distinct analytic germs would clarify the geometry of the solution set near infinity.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies autonomous first-order algebraic ordinary differential equations F(y,y')=0 over the complex numbers and claims that every formal Puiseux series solution, expanded at any finite point or at infinity, is convergent. The central device is a map Δ associating to a formal Puiseux solution of ramification order n the formal parametrization (a(t),b(t))=(y(t^n), t^{hn}y'(t^n)) of the curve F(y,p)=0, so that solution places are characterized by the associated first-order differential equation a'(s)s'=n t^{n(1-h)-1}b(s). The authors prove the main convergence statement in Theorem 3.11 via a Briot-Bouquet lemma, give an existence theorem for analytic solutions through arbitrary points in Theorem 3.12, and provide algorithms, with truncation bounds, for computing all Puiseux series solutions around zero and at infinity.

Significance. If the main theorem is correct, it is a substantial extension of the authors' earlier power-series result [9] to fractional-power solutions: formal Puiseux solutions of autonomous first-order algebraic ODEs would never be merely formal objects, and would always correspond to analytic solutions of the equation. The proof is constructive, self-contained modulo classical Puiseux parametrization and the Briot-Bouquet theorem, and the paper gives algorithmic descriptions with explicit truncation bounds in the finite-point case. The existence theorem for an analytic solution through every point is a clean by-product. The main gaps I found are the handling of the sign in the infinity reduction of Theorem 3.11 and an off-by-one indexing error in Lemma 3.9(2); both are local and repairable, but they need to be fixed before the paper can be accepted.

major comments (2)
  1. [Theorem 3.11 and Section 4.2] The proof of the infinity half of Theorem 3.11 is incomplete as written. The reduction x=1/z in Section 2 turns a solution around infinity of F(y,y')=0 into a solution of F(y(z), -z^2 y'(z))=0, but all results in Section 3, in particular Lemma 3.2 and Lemma 3.9 with h=2, are proved for the plus-sign equation (2.2), F(y,x^h y')=0. Section 4.2 acknowledges the sign change only by saying 'up to the sign' and by writing the different associated equation (4.1), but no version of Lemma 3.9 or Theorem 3.11 is proved for (4.1), and Corollary 4.4 invokes Lemma 3.9 for (4.1) without justification. Example 4.5 shows that the distinction is not cosmetic: the plus-sign equation can fail to have a family where the true infinity equation has one. This gap is easily closed by applying the plus-sign theory to the reflected polynomial \tilde F(y,p)=F(y,-p), since the infinity equation for F is exactly the plus-sign equation (2.2) with h=2 for \tilde F. That reduction must be stated and verified explicitly; without it, the convergence proof for expansions at infinity does not follow from the displayed argument.
  2. [Lemma 3.9(2)] The free parameter in Lemma 3.9(2) is misindexed. For h≥2, ν=n(h-1)=r-k>0, and the proof's change of variables s(t)=t(σ+z(t)) leads to λ=ν in Lemma 3.8. Lemma 3.8(2) makes ζ_ν the free coefficient of z(t), hence σ_{ν+1}=σ_{r-k+1} is the free coefficient of s(t), not σ_{r-k} as stated. The proof itself contains the correct index, but the statement and its use in the parameter-counting parts of Section 4.2 need to be corrected.
minor comments (4)
  1. [Lemma 3.8(2)(a)] The convergence statement in the resonant case λ∈Z_{>0} is dispatched with a one-sentence reference to Section 86 of [4]. Since Theorem 3.11 relies on this case, please either spell out the reduction to the non-resonant case or state the precise classical theorem invoked.
  2. [Example 4.5] The example closes by saying that the displayed family describes all formal Puiseux series solutions expanded around infinity, but the constant solution y=0 is not included in the family. Please clarify that the family describes all non-constant solutions, or state explicitly that the constant solution must be added.
  3. [Theorem 4.1] The truncated reparametrizations are written as \hat s_i(t)=\sum_{j=1}^H σ_{i,j} t^j, but the integer H is not defined in the proof; it should presumably be N or another explicitly introduced bound.
  4. [Section 4.2] The phrase 'up to the sign' is too informal for a proof. If the sign issue is resolved by passing to \tilde F(y,p)=F(y,-p), the reduction should appear in Section 2 or at the start of Section 4.2 rather than as a parenthetical remark.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof reduces formal Puiseux solutions to place parametrizations and the classical Briot–Bouquet theorem, with no fitted input or self-citation chain carrying the claim.

full rationale

The derivation is self-contained with respect to external, non-circular support. Theorem 3.11 is proved via the place construction: a solution y(x) maps to the parametrization (a(t), b(t)) satisfying a'(t) = n t^{n(1-h)-1} b(t) (Lemma 3.2), and the key reparametrization step is reduced to Lemma 3.9, which in turn is reduced to the classical Briot–Bouquet Lemma 3.8. Lemma 3.8's existence and uniqueness are proved in the paper itself, while its convergence assertions rest on the 1856 external source [4], not on the authors' own prior work. The only self-citation, [9] (Falkensteiner–Sendra), marks the previously known power-series case and supplies a comparison example (Example 4.2); Theorem 3.11 does not invoke [9] as a premise and instead reproves convergence through place parametrizations and Puiseux's theorem. Theorem 3.12 uses Puiseux's theorem and Theorem 3.10, again independent algebraic facts. No fitted parameter is renamed as a prediction, and no equation is assumed in the form being derived. The skeptic's infinity-sign objection is a proof-completeness or correctness concern, not circularity: the gap is between the two genuinely different equations F(y, -z^2 y')=0 and F(y, z^2 y')=0, not an identification of the theorem's conclusion with one of its hypotheses.

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

No free parameters are fitted to data; the paper is a pure proof. It relies on standard algebraic-curve facts and the classical Briot-Bouquet theorem, with one delegation to the literature for the resonant convergence step. It introduces no new postulated entities, particles, fields, or dimensions.

assumptions (4)
  • standard math Puiseux parametrizations of algebraic curves are convergent (Puiseux's theorem).
    Used in Theorem 3.11 to conclude that the curve parametrization (t^k, b_bar(t)) is convergent, and in Theorem 3.12 to obtain a convergent p(y). This is classical algebraic curve theory, not proved in the paper.
  • standard math For every place of an algebraic curve there is a formal reparametrization s(t) of order one with a(s(t))-y0=t^k (Walker, Chapter IV, Section 2).
    Used at the start of the proof of Theorem 3.11; a standard fact in place theory that the paper invokes without proof.
  • domain assumption In the resonant case lambda in Z_{>0}, formal solutions of the Briot-Bouquet equation are convergent, via a change of variables reducing to the non-resonant case.
    Invoked in Lemma 3.8(2)(a) and needed for the h>=2 infinity case; the paper cites Section 86 of [4] rather than proving the reduction in detail.
  • standard math Bounds on truncations of places from Duval [8] and Stadelmeyer [16] ensure that the algorithms cover all places and terminate.
    Used in Algorithms 1 and 2 for correctness and termination; taken from the cited literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Existence and convergence of Puiseux series solutions for autonomous first order differential equations." pith.science (2026). https://pith.science/paper/BBLKRZRJ

@misc{pith2026190809196,
  author       = {Pith},
  title        = {Pith review of: Existence and convergence of Puiseux series solutions for autonomous first order differential equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BBLKRZRJ}},
  note         = {Machine review of arXiv:1908.09196}
}
read the original abstract

Given an autonomous first order algebraic ordinary differential equation F(y,y')=0, we prove that every formal Puiseux series solution, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.

Figures

Figures reproduced from arXiv: 1908.09196 by the authors.

Figure 1
Figure 1. The Newton polygon of the algebraic curve F(y, p) = 0. All its sides have non-negative slope, be￾cause the point (0, 0) ∈ N (F). Since the degree of F(y, p) with respect to p is positive, N (F) has at least one side. Therefore, by Puiseux’s Theorem, there exists a conver￾gent Puiseux series solution p(y) of the algebraic equation F(y, p(y)) = 0 of the form p(y) = P i≥k ci y i/n, where ck 6= 0 and k ≤ 0. Let us defin… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [9]

    Falkensteiner, S., & Sendra, J.R. 2019. Formal Power Series So- lutions of First Order Autonomous Algebraic Ordinary Differential Equations. Mathematics in Computer Science

    work page 2019

  2. [1]

    Aroca, J.M. 2000. Puiseux Solutions of Singular Differential Equa- tions. Basel: Birkh¨ auser Basel. Pages 129–145

    work page 2000

  3. [2]

    Aroca, J.M., Cano, J., Feng, R., & Gao, X.-S. 2005. Algebraic gen- eral solutions of algebraic ordinary differential equations. Pages 29–36 of: Proceedings of the 2005 international symposium on Symb olic and algebraic computation. ACM

    work page 2005

  4. [3]

    Aroca Bisquert, Fuensanta. 2000. M´ etodos algebraicos en ecua- ciones diferenciales ordinarias en el campo complejo . Ph.D. thesis, Univ. de Valladolid

    work page 2000

  5. [4]

    Briot, C.A., & Bouquet, J.C. 1856. Recherches sur les propriet´ e s des ´ equations diff´ erentielles.Journal de l’Ecole Polytechnique , 21:36, 133–198. 22 JOS ´E CANO, SEBASTIAN F ALKENSTEINER, AND J.RAF AEL SENDRA

    work page

  6. [5]

    Cano, J. 1993. An extension of the Newton-Puiseux polygon con - struction to give solutions of Pfaffian forms. Annales de l’Institut Fourier, 43(1), 125–142

    work page 1993

  7. [6]

    Cano, J. 2005. The Newton Polygon Method for Differential Equa - tions. Pages 18–30 of: Proceedings of the 6th International Confer - ence on Computer Algebra and Geometric Algebra with Applica tions. IWMM’04/GIAE’04. Berlin, Heidelberg: Springer-Verlag

    work page 2005

  8. [7]

    Della Dora, J., & Richard-Jung, F. 1997. About the Newton Algo- rithm for Non-linear Ordinary Differential Equations. Pages 298–304 of: Proceedings of the 1997 International Symposium on Symb olic and Algebraic Computation. ISSAC ’97. New York, NY, USA: ACM

    work page 1997

Show all 18 references
  1. [8]

    Duval, D. 1989. Rational Puiseux expansion. Compositio Mathe- matica, 70(2), 119–154

    1989

  2. [10]

    Feng, R., & Gao, X.-S. 2004. Rational general solutions of alge- braic ordinary differential equations. Pages 155–162 of: Proceedings of the 2004 international symposium on Symbolic and algebra ic com- putation. ACM

    2004

  3. [11]

    Feng, R., & Gao, X.-S. 2006. A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs . Journal of Symbolic computation , 41(7), 739–762

    2006

  4. [12]

    Fine, H. 1889. On the Functions defined by Differential Equation s, with an Extension of the Puiseux Polygon Construction to these Equ a- tions. American Journal of Mathematics , 11, 317–328

  5. [13]

    Fine, H. 1890. Singular Solutions of Ordinary Differential Equa- tions. American Journal of Mathematics , 12, 295–322

  6. [14]

    Grigoriev, D.Y., & Singer, M. 1991. Solving Ordinary Differential Equations in Terms of Series with Real Exponents. Trans A.M.S. , 327, 329–351

    1991

  7. [15]

    Ince, E. 1926. Ordinary Differential Equations . Dover

    1926

  8. [16]

    Stadelmeyer, P. 2000. On the Computational Complexity of Re- solving Curve Singularities and Related Problems . Ph.D. thesis, RISC, Johannes Kepler University Linz

    2000

  9. [17]

    Vo, N.T., Grasegger, G., & Winkler, F. 2018. Deciding the Exis- tence of Rational General Solutions for First-Order Algebraic ODE s. Journal of Symbolic Computation , 87, 127–139

    2018

  10. [18]

    Walker, R.J. 1950. Algebraic Curves. Princeton University Press. EXISTENCE AND CONVERGENCE OF SERIES SOLUTIONS OF ODE’S 23 Dpto. Algebra, an ´alisis matem ´atico, geometr ´ıa y topolog ´ıa, Uni- versidad de V alladolid, Spain. E-mail address : jcano@agt.uva.es Research Institu...

    1950

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.