Elementary solutions of ordinary tropical differential equations, and vanishing orders of solutions of algebraic differential equations

Carla Valencia-Negrete; Cristhian Garay-L\'opez; Johana Luviano-Flores

arxiv: 2606.21829 · v1 · pith:KLJ67UF4new · submitted 2026-06-20 · 🧮 math.AG · math.CO

Elementary solutions of ordinary tropical differential equations, and vanishing orders of solutions of algebraic differential equations

Cristhian Garay-L\'opez , Johana Luviano-Flores , Carla Valencia-Negrete This is my paper

Pith reviewed 2026-06-26 11:43 UTC · model grok-4.3

classification 🧮 math.AG math.CO

keywords tropical differential equationsalgebraic differential equationsformal power series solutionsvanishing ordersboolean semifieldtropicalizationHahn seriesordinary differential equations

0 comments

The pith

The t-adic orders of formal Hahn solutions to an algebraic ODE are contained in the orders of elementary k-solutions to its tropicalization.

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

The paper shows that for an ordinary algebraic differential equation 𝔓 of order k with meromorphic coefficients, the set S(𝔓,0) of t-adic orders of its formal Hahn series solutions over ℂ[[t^ℝ]] sits inside the set of t-adic orders of elementary k-solutions to the tropical equation P = trop(𝔓) over the boolean semifield ℬ[[t^ℝ]]. It first develops the tropical side by comparing the sets of elementary k-solutions and minimal tropical solutions for supports in different submonoids Γ of ℝ, establishing that these sets behave similarly. The containment then follows by relating the algebraic solutions to these tropical ones. A reader cares because the result supplies a combinatorial, effectively computable description of possible vanishing orders at t=0 using only tropical algebra.

Core claim

Given an ordinary algebraic differential equation 𝔓(y) of differential order k, the set S(𝔓,0) := ord_t(Sol_ℂ[[t^ℝ]](𝔓)) is contained in ord_t(Sol_ℬ[[t^ℝ]],k(P)) where P = trop(𝔓). In most cases the right-hand set is realized by a finite family of univariate tropical polynomials.

What carries the argument

The inclusion S(𝔓,0) ⊂ ord_t(Sol_ℬ[[t^ℝ]],k(P)) realized by the correspondence between t-adic orders of algebraic Hahn solutions and those of elementary k-solutions of the tropicalized equation over the boolean semifield.

If this is right

The sets Sol_ℬ[[t^Γ]],k(P) and μ(Sol_ℬ[[t^Γ]](P)) of elementary k-solutions and minimal tropical solutions share many structural similarities across choices of support monoid Γ.
For most tropical equations the set Sol_ℬ[[t^ℝ]],k(P) consists of a finite collection of univariate tropical polynomials.
The containment supplies combinatorial information on the nature of germs of solutions of 𝔓 at t=0.
Tropical algebra over ℬ yields an effective method to locate possible orders without solving the original algebraic equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The finite character of the tropical solution set in most cases suggests an algorithm that enumerates candidate orders by solving low-degree tropical polynomials.
The same containment may furnish bounds on solution orders for equations whose coefficients lie in larger fields once a suitable tropicalization is defined.
The approach could be tested on concrete examples such as Riccati or Painlevé equations to see whether the tropical orders match observed asymptotic behaviors.

Load-bearing premise

The t-adic orders of actual complex solutions are always realized as orders of elementary k-solutions of the tropicalized equation over the boolean semifield.

What would settle it

An algebraic differential equation 𝔓 whose formal Hahn solution has a t-adic order absent from every elementary k-solution of P = trop(𝔓).

read the original abstract

Our aim is to use tropical differential algebra to systematically build a combinatorial basis for the study of the set of (formal) power series solutions to nonlinear algebraic ordinary differential equations (over $\mathbb{C}$) expanded around the point $t_0=0\in\mathbb{C}$, which may also be effectively computed using standard tropical algebra. This paper is divided into two parts. First, given an ordinary tropical differential equation in one differential variable $P=P(y)$ of (differential) order $k$, we study the sets $Sol_{\mathbb{B}[\![t^{\Gamma}]\!],k}(P)\supset \mu(Sol_{\mathbb{B}[\![t^{\Gamma}]\!]}(P)\!)$ of tropical elementary $k$-solutions and minimal tropical solutions, respectively; we show that these two sets bear many similarities. We do this for tropical solutions $y=\varphi(t)\in \mathbb{B}[\![t^{\Gamma}]\!]$ (with coefficients in the boolean semifield $\mathbb{B}$) of $P$ having support in different relevant submonoids $\Gamma$ of $(\mathbb{R},+,0)$. Then, given an ordinary algebraic differential equation $\mathfrak{P}$ (with meromorphic coefficients in one differential variable $\mathfrak{P}=\mathfrak{P}(y)$ and of differential order $k$), we consider the set $S(\mathfrak{P},0):=ord_t(Sol_{\mathbb{C}[\![t^{\mathbb{R}}]\!]}(\mathfrak{P})\!)\subset\mathbb{R}$ of $t$-adic orders of formal Hahn solutions of $\mathfrak{P}$, which is an algebraic object that gives information about the nature of the germs of solutions of $\mathfrak{P}$ at the point $t_0=0\in\mathbb{C}$. We show that this set is contained in the set of $t$-adic orders of Hahn elementary $k$-solutions of its tropicalization $P=trop(\mathfrak{P})$, this is $S(\mathfrak{P},0)\subset ord_t(Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P))$. In most cases, the set $Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P)$ is a finite family of univariate tropical polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a concrete inclusion linking t-adic orders of algebraic formal solutions to orders of elementary tropical k-solutions of the tropicalized equation.

read the letter

The central result is the inclusion S(𝔓,0) ⊂ ord_t(Sol_𝔹[[t^ℝ]],k(P)) for P = trop(𝔓). They also define tropical elementary k-solutions and minimal tropical solutions over the boolean semifield on various supports Γ, and observe that these two sets behave similarly. In most cases the tropical solution set is finite, which is a practical feature.

This is useful work if you already care about tropical differential algebra. It turns the algebraic vanishing-order question into a combinatorial one that can be checked with standard tropical operations, and the setup with Hahn series and different monoids Γ looks deliberate. The claim is stated cleanly and the direction of the inclusion matches what one would hope for from a valuation-preserving tropicalization.

The main limitation is that the abstract supplies the statement but no proof outline or key lemma, so it is impossible to judge from the given text whether the argument closes or whether the correspondence holds only for restricted supports. The scope stays strictly local and formal, so it does not speak to convergence or global behavior. No obvious circularity or internal contradiction appears in the structure.

This is for readers already working in tropical methods for differential equations or formal series solutions. Outside that group the payoff is small. The paper deserves a serious referee: the claim is specific enough to be checked and the definitions are new enough to matter inside the subfield.

Referee Report

0 major / 3 minor

Summary. The paper develops tropical differential algebra to study formal power series solutions of nonlinear algebraic ODEs over C expanded at t=0. Part 1 examines, for a tropical ODE P(y) of order k, the sets Sol_B[[t^Γ]],k(P) of elementary k-solutions and the minimal solutions μ(Sol_B[[t^Γ]](P)) over the boolean semifield for various supports Γ, establishing similarities between them. Part 2 defines S(𝔓,0) as the set of t-adic orders of formal Hahn solutions to an algebraic ODE 𝔓 of order k, and proves the inclusion S(𝔓,0) ⊂ ord_t(Sol_B[[t^R]],k(P)) with P = trop(𝔓); in most cases the right-hand set is finite.

Significance. If the inclusion holds, the result supplies a combinatorial, often finite, description of possible vanishing orders of algebraic solutions via tropical elementary solutions, directly linking algebraic and tropical differential equations and enabling effective computation of bounds on solution germs at t=0. The explicit construction over multiple supports Γ and the finiteness statement for Sol_B[[t^R]],k(P) are concrete strengths.

minor comments (3)

[Abstract and §1] The abstract and introduction use both 'Hahn elementary k-solutions' and 'tropical elementary k-solutions' for the same objects; a single consistent term would improve readability.
[§2] Notation for the support monoids alternates between Γ and R without an explicit comparison table; adding a short table in §2 listing the Γ considered and the corresponding solution sets would clarify the scope of the similarities proved.
[§4] The statement that Sol_B[[t^R]],k(P) is 'in most cases' a finite family of univariate tropical polynomials is repeated but not quantified; a precise criterion (e.g., in terms of the Newton polygon or the differential order k) would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for recognizing the significance of the inclusion relating algebraic solution orders to tropical elementary solutions, and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines S(𝔓,0) as the set of t-adic orders of formal Hahn solutions to the algebraic ODE 𝔓 over ℂ[[t^ℝ]], and separately defines the tropical elementary k-solutions over the boolean semifield 𝔹[[t^Γ]] for the tropicalized equation P = trop(𝔓). The claimed result is an inclusion between these two independently constructed sets. The abstract and structure present the tropical analysis first as an independent combinatorial study, followed by a proof of containment; no equation or step equates one side to the other by definition, renames a fitted quantity as a prediction, or reduces the central claim to a self-citation chain. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract invokes standard tropical algebra and the boolean semifield without listing new free parameters or invented entities; the correspondence between algebraic and tropical orders is the central unproven link.

pith-pipeline@v0.9.1-grok · 5971 in / 1164 out tokens · 27666 ms · 2026-06-26T11:43:16.168836+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 7 canonical work pages

[1]

J., Fokas, A

Ablowitz, M. J., Fokas, A. S. (2003).Complex variables: introduction and applications(2nd ed). Cambridge University Press

2003
[2]

Ahlfors, L. V. (1979).Complex analysis: An introduction to the theory of analytic functions of one complex variable(3rd ed.). McGraw-Hill

1979
[3]

Aroca, F., Bossinger, L., Falkensteiner, S., Garay, C., et al.:Infinite matroids in tropical differential algebra. Can. Math. Bull. 2026;69(1):201–221. https://doi.org/10.4153/S0008439525101094

work page doi:10.4153/s0008439525101094 2026
[4]

Pacific J

Aroca, F., Garay, C., Toghani, Z.The fundamental theorem of tropical differential algebraic geometry. Pacific J. Math., 283(2):257–270, 2016

2016
[5]

W., Churchill, R

Brown, J. W., Churchill, R. V. (2009).Complex Variables and Applications(8th ed.). McGraw-Hill

2009
[6]

An algebraic and geometric approach

Campillo, A., Castellanos, J.Curve singularities. An algebraic and geometric approach. Hermann, 2005

2005
[7]

Conte and M

Conte, R., Musette, M. (2020).The Painlev´ e handbook(2nd ed.). Springer Nature. https://doi.org/10.1007/978-3-030-53340-3

work page doi:10.1007/978-3-030-53340-3 2020
[8]

Cotterill, E., Garay, C., Luviano, J.Exploring tropical differential equationsAdvances in Geometry, vol. 23, no. 4, 2023, pp. 437-460. https://doi.org/10.1515/advgeom-2023-0019

work page doi:10.1515/advgeom-2023-0019 2023
[9]

Results Math 77, 41 (2022)

Filipuk, G., Kecker, T.On Singularities of Certain Non-linear Second-Order Ordinary Differential Equa- tions. Results Math 77, 41 (2022). https://doi.org/10.1007/s00025-021-01577-1

work page doi:10.1007/s00025-021-01577-1 2022
[10]

Annales scientifiques de l’ ´Ecole Normale Sup´ erieure, Serie 3, Volume 29 (1912), pp

Garnier, R.Sur des ´ equations diff´ erentielles du troisi` eme ordre dont l’int´ egrale g´ en´ erale est uniforme et sur une classe d’´ equations nouvelles d’ordre sup´ erieur dont l’int´ egrale g´ en´ erale a ses points critiques fixes. Annales scientifiques de l’ ´Ecole Normale Sup´ erieure, Serie 3, Volume 29 (1912), pp. 1-126. doi: 10.24033/asens.644

work page doi:10.24033/asens.644 1912
[11]

Advances in Applied Mathematics Volume 82, pp

Grigoriev, D.Tropical differential equations. Advances in Applied Mathematics Volume 82, pp. 120-128
[12]

and Garay, C.Minimal solutions of tropical linear differential systems

Grigoriev, D. and Garay, C.Minimal solutions of tropical linear differential systems. AAECC (2026). https://doi.org/10.1007/s00200-025-00722-5

work page doi:10.1007/s00200-025-00722-5 2026
[13]

and Singer, M

Grigoriev, D. and Singer, M. F.Solving ordinary differential equations in terms of series with real expo- nents. Transactions of the AMS, Vol. 327, No. 1, pp. 329-351
[14]

Joshi, N., Kruskal, M. D. (1994).A direct proof that the solutions of the six Painlev´ e equations have no movable singularities except poles. Studies in Applied Mathematics, 93(3), 187–207

1994
[15]

and Sturmfels, B.Introduction to Tropical Geometry, American Mathematical Society, Grad- uate Studies in Mathematics

Maclagan, D. and Sturmfels, B.Introduction to Tropical Geometry, American Mathematical Society, Grad- uate Studies in Mathematics. Volume: 161; 2015

2015
[16]

and Gallinaro, F.The Fundamental theorem of tropical differential algebra over nontriv- ially valued fields and the radius of convergence of nonarchimedean differential equations

Mereta, S. and Gallinaro, F.The Fundamental theorem of tropical differential algebra over nontriv- ially valued fields and the radius of convergence of nonarchimedean differential equations. Preprint ArXiv:2303.12124v3 [math.AG]

Pith/arXiv arXiv
[17]

Bulletin de la Soci´ et´ e Math´ ematique de France, Volume 28 (1900), pp

Painlev´ e, P.M´ emoire sur les ´ equations diff´ erentielles dont l’int´ egrale g´ en´ erale est uniforme. Bulletin de la Soci´ et´ e Math´ ematique de France, Volume 28 (1900), pp. 201-261. doi: 10.24033/bsmf.633

work page doi:10.24033/bsmf.633 1900
[18]

(1989).The Painlev´ e property and singularity analysis of integrable and non-integrable systems

Ramani, A., Grammaticos, B., Bountis, T. (1989).The Painlev´ e property and singularity analysis of integrable and non-integrable systems. Physics Reports, 180, 159-245. Centro de Investigaci ´on en Matem´aticas, A.C. (CIMAT) Jalisco S/N, Col. Valenciana CP. 36023 Guanajuato, Gto, M´exico Email address:cristhian.garay@cimat.mx Universidad Aut´onoma Metrop...

1989

[1] [1]

J., Fokas, A

Ablowitz, M. J., Fokas, A. S. (2003).Complex variables: introduction and applications(2nd ed). Cambridge University Press

2003

[2] [2]

Ahlfors, L. V. (1979).Complex analysis: An introduction to the theory of analytic functions of one complex variable(3rd ed.). McGraw-Hill

1979

[3] [3]

Aroca, F., Bossinger, L., Falkensteiner, S., Garay, C., et al.:Infinite matroids in tropical differential algebra. Can. Math. Bull. 2026;69(1):201–221. https://doi.org/10.4153/S0008439525101094

work page doi:10.4153/s0008439525101094 2026

[4] [4]

Pacific J

Aroca, F., Garay, C., Toghani, Z.The fundamental theorem of tropical differential algebraic geometry. Pacific J. Math., 283(2):257–270, 2016

2016

[5] [5]

W., Churchill, R

Brown, J. W., Churchill, R. V. (2009).Complex Variables and Applications(8th ed.). McGraw-Hill

2009

[6] [6]

An algebraic and geometric approach

Campillo, A., Castellanos, J.Curve singularities. An algebraic and geometric approach. Hermann, 2005

2005

[7] [7]

Conte and M

Conte, R., Musette, M. (2020).The Painlev´ e handbook(2nd ed.). Springer Nature. https://doi.org/10.1007/978-3-030-53340-3

work page doi:10.1007/978-3-030-53340-3 2020

[8] [8]

Cotterill, E., Garay, C., Luviano, J.Exploring tropical differential equationsAdvances in Geometry, vol. 23, no. 4, 2023, pp. 437-460. https://doi.org/10.1515/advgeom-2023-0019

work page doi:10.1515/advgeom-2023-0019 2023

[9] [9]

Results Math 77, 41 (2022)

Filipuk, G., Kecker, T.On Singularities of Certain Non-linear Second-Order Ordinary Differential Equa- tions. Results Math 77, 41 (2022). https://doi.org/10.1007/s00025-021-01577-1

work page doi:10.1007/s00025-021-01577-1 2022

[10] [10]

Annales scientifiques de l’ ´Ecole Normale Sup´ erieure, Serie 3, Volume 29 (1912), pp

Garnier, R.Sur des ´ equations diff´ erentielles du troisi` eme ordre dont l’int´ egrale g´ en´ erale est uniforme et sur une classe d’´ equations nouvelles d’ordre sup´ erieur dont l’int´ egrale g´ en´ erale a ses points critiques fixes. Annales scientifiques de l’ ´Ecole Normale Sup´ erieure, Serie 3, Volume 29 (1912), pp. 1-126. doi: 10.24033/asens.644

work page doi:10.24033/asens.644 1912

[11] [11]

Advances in Applied Mathematics Volume 82, pp

Grigoriev, D.Tropical differential equations. Advances in Applied Mathematics Volume 82, pp. 120-128

[12] [12]

and Garay, C.Minimal solutions of tropical linear differential systems

Grigoriev, D. and Garay, C.Minimal solutions of tropical linear differential systems. AAECC (2026). https://doi.org/10.1007/s00200-025-00722-5

work page doi:10.1007/s00200-025-00722-5 2026

[13] [13]

and Singer, M

Grigoriev, D. and Singer, M. F.Solving ordinary differential equations in terms of series with real expo- nents. Transactions of the AMS, Vol. 327, No. 1, pp. 329-351

[14] [14]

Joshi, N., Kruskal, M. D. (1994).A direct proof that the solutions of the six Painlev´ e equations have no movable singularities except poles. Studies in Applied Mathematics, 93(3), 187–207

1994

[15] [15]

and Sturmfels, B.Introduction to Tropical Geometry, American Mathematical Society, Grad- uate Studies in Mathematics

Maclagan, D. and Sturmfels, B.Introduction to Tropical Geometry, American Mathematical Society, Grad- uate Studies in Mathematics. Volume: 161; 2015

2015

[16] [16]

and Gallinaro, F.The Fundamental theorem of tropical differential algebra over nontriv- ially valued fields and the radius of convergence of nonarchimedean differential equations

Mereta, S. and Gallinaro, F.The Fundamental theorem of tropical differential algebra over nontriv- ially valued fields and the radius of convergence of nonarchimedean differential equations. Preprint ArXiv:2303.12124v3 [math.AG]

Pith/arXiv arXiv

[17] [17]

Bulletin de la Soci´ et´ e Math´ ematique de France, Volume 28 (1900), pp

Painlev´ e, P.M´ emoire sur les ´ equations diff´ erentielles dont l’int´ egrale g´ en´ erale est uniforme. Bulletin de la Soci´ et´ e Math´ ematique de France, Volume 28 (1900), pp. 201-261. doi: 10.24033/bsmf.633

work page doi:10.24033/bsmf.633 1900

[18] [18]

(1989).The Painlev´ e property and singularity analysis of integrable and non-integrable systems

Ramani, A., Grammaticos, B., Bountis, T. (1989).The Painlev´ e property and singularity analysis of integrable and non-integrable systems. Physics Reports, 180, 159-245. Centro de Investigaci ´on en Matem´aticas, A.C. (CIMAT) Jalisco S/N, Col. Valenciana CP. 36023 Guanajuato, Gto, M´exico Email address:cristhian.garay@cimat.mx Universidad Aut´onoma Metrop...

1989