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Evaluating rational functions over the tropical semiring converts algebraic identities into combinatorial statements that serve as proofs.

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2026-06-29 06:11 UTC pith:OO2EO27L

load-bearing objection The paper shows detropicalization via tropical semiring evaluation works as a proof technique for the subtraction-free rational function identities in its examples, though the contribution stays illustrative.

arxiv 2605.30511 v1 pith:OO2EO27L submitted 2026-05-28 math.CO

Detropicalization as a proof technique

Darij Grinberg , Tom Roby This is my paper
classification math.CO
keywords tropical semiringdetropicalizationcombinatorial proofsrational functionssemiringsalgebraic identitiesmax-plus algebra
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that rational functions free of subtraction remain well-defined over semirings and acquire direct combinatorial interpretations when evaluated in the tropical semiring, where addition is replaced by maximum and multiplication by ordinary addition. These interpretations supply combinatorial proofs of the original identities, which can then be lifted back because the semiring evaluation preserves the relevant structure. A reader would care because the method turns algebraic verification into counting arguments without requiring subtraction or field inverses. The approach is demonstrated through concrete examples rather than a single general theorem.

Core claim

Rational functions make sense over any commutative ring with invertible denominators, and when subtraction is absent they extend to semirings. Evaluating them in the tropical semiring replaces addition by max and multiplication by addition, so that algebraic identities acquire combinatorial meaning; this meaning can be used to prove the identities, after which the proofs are detropicalized back to the original setting.

What carries the argument

Detropicalization: the process of reading a combinatorial identity obtained from a tropical (max-plus) evaluation and lifting it to a proof of the corresponding algebraic identity over rings or semirings.

Load-bearing premise

The tropical evaluation must preserve enough structure so that a true combinatorial identity in the max-plus semiring implies the original algebraic identity holds.

What would settle it

An explicit rational function without subtraction for which the tropical max-plus version yields a valid combinatorial identity but the original identity fails over the integers or rationals.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Identities involving only positive terms in generating functions or polynomials can be proved by counting objects whose sizes are governed by max and plus operations.
  • Any algebraic relation that remains defined after replacing plus with max and times with plus becomes a candidate for a combinatorial proof via this route.
  • The method applies directly to identities in which every denominator is a monomial, since such expressions stay meaningful in semirings.

Where Pith is reading between the lines

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

  • The technique may extend to identities in noncommutative settings if a suitable noncommutative tropical semiring can be defined.
  • It offers a systematic way to discover new combinatorial interpretations for known algebraic identities by first tropicalizing them.
  • Applications could include transfer-matrix identities or recursions where the absence of subtraction is already natural.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The manuscript proposes detropicalization as a proof technique: for rational functions over commutative rings that involve no subtraction (hence well-defined over semirings), evaluation in the tropical semiring (max-plus) yields combinatorial statements whose validity can be lifted back to prove the original algebraic identities.

Significance. If the lifting step is valid, the method supplies a systematic route from algebraic identities to combinatorial proofs via tropical evaluation, with the supplied examples serving as concrete, checkable instances. This adds a potentially useful bridge between algebra and combinatorics when the no-subtraction hypothesis holds.

minor comments (2)
  1. The manuscript would benefit from an explicit statement, perhaps in the introduction, of the precise conditions under which the combinatorial identity obtained tropically lifts to the ring setting (e.g., a short lemma on preservation of the relevant operations).
  2. Notation for the tropical semiring operations should be introduced once at the beginning rather than assumed from context in the examples.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the accurate summary of the detropicalization technique, and the recommendation to accept. The assessment correctly identifies the core hypothesis (no-subtraction rational functions) and the potential bridge between algebra and combinatorics.

Circularity Check

0 steps flagged

No significant circularity; technique is self-contained

full rationale

The paper presents a proof technique of evaluating subtraction-free rational functions over the tropical semiring to obtain combinatorial identities that are then lifted back to the original ring setting. No equations, parameters, or self-citations are used in a load-bearing way that reduces the central claim to its own inputs by construction. The argument relies on the semiring axioms and the explicit no-subtraction hypothesis, with examples serving as direct verification rather than fitted predictions or renamed prior results. This matches the default expectation of a non-circular paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms or invented entities are stated. The implicit domain assumption is that the rational functions contain no subtraction.

axioms (1)
  • domain assumption Rational functions without subtraction are well-defined over semirings
    Stated in the first two sentences of the abstract

pith-pipeline@v0.9.1-grok · 5579 in / 1060 out tokens · 17462 ms · 2026-06-29T06:11:54.409578+00:00 · methodology

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Cite this review

Pith. "Pith review of Detropicalization as a proof technique." pith.science (2026). https://pith.science/paper/OO2EO27L

@misc{pith2026260530511,
  author       = {Pith},
  title        = {Pith review of: Detropicalization as a proof technique},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OO2EO27L}},
  note         = {Machine review of arXiv:2605.30511}
}
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read the original abstract

Rational functions make sense over any commutative ring, as long as the denominators are invertible. When there is no subtraction involved, they even apply over semirings (rings without subtraction). It is particularly worthwhile to evaluate them over the *tropical semiring*, in which the roles of addition and multiplication are played by maxima and addition. Over this semiring, algebraic results often acquire combinatorial meaning. We give a few examples.

Figures

Figures reproduced from arXiv: 2605.30511 by Darij Grinberg, Tom Roby.

Figure 1
Figure 1. Figure 1: Example for the racetrack puzzle (n = 5). The five gas cans A, B, C, D, E have respectively 0.2, 0.1, 0.4, 0.2, 0.1 gallons of gas (num￾bers in parentheses). The numbers on the arcs show how much gas is needed to go from one can to the next. An example is shown on [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A rowmotion family for p = 2 and q = 2. The values xi,j,k are shown for all 0 ≤ k ≤ 4 and all i, j. The k-coordinate in xi,j,k increases in the upwards direction; the other coordinates are oblique. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

33 extracted references · 7 canonical work pages · 1 internal anchor

  1. [1]

    Diverse Autoren,Schweizer Mathematikolympiade: Alte Prüfungen, https: //mathematical.olympiad.ch/de/alte-pruefungen

  2. [2]

    Véronique Bazier-Matte,Quasi-cluster algebras: An overview, Representa- tions of Algebras and Related Topics: Proceedings of the Workshop and the 20th International Conference on Representations of Algebras, ICRA 2022, Montevideo, Uruguay and Buenos Aires, Argentina, 3–12 August 2022, EMS, 2025

    2022

  3. [3]

    Marie Charlotte Brandenburg, Georg Loho, and Guido Montúfar,The Real Tropical Geometry of Neural Networks, Preprint, arXiv:2403.11871 [math.CO] (2024), 2024, https://arxiv.org/abs/2403.11871

    work page arXiv 2024

  4. [4]

    Erwan Brugallé,Some aspects of tropical geometry, Eur. Math. Soc. Newsl. 83(2012), 23–28,ISSN 1027-488X, http://erwan.brugalle.perso.math.cnrs. fr/articles/EMS/TropEMS.pdf

    2012

  5. [5]

    Erwan Brugallé, Ilia Itenberg, Kristin Shaw, and Oleg Viro,Tropical geometry, Snapshots of modern mathematics from Oberwolfach (2018), https://doi.org/10.14760/SNAP-2018-007-EN

    work page doi:10.14760/snap-2018-007-en 2018

  6. [6]

    Representa- tions and combinatorics, Hackensack, NJ: World Scientific, 2017, ISBN 978-981-4733-43-4; 978-981-4733-44-1

    Daniel Bump and Anne Schilling,Crystal bases. Representa- tions and combinatorics, Hackensack, NJ: World Scientific, 2017, ISBN 978-981-4733-43-4; 978-981-4733-44-1

    2017

  7. [7]

    Comb.4(2021), no

    David Einstein and James Propp,Combinatorial, piecewise-linear, and birational homomesy for products of two chains, Algebr. Comb.4(2021), no. 2, 201–224, ISSN 2589-5486

    2021

  8. [8]

    Fordy and Bethany R

    Allan P. Fordy and Bethany R. Marsh,Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebr. Comb.34(2011), no. 1, 19–66, ISSN 0925-9899

    2011

  9. [9]

    Jeffrey Giansiracusa, Kevin Kuehn, Stefano Mereta, and Eduardo Vital, Three lectures on tropical algebra, 2026, https://arxiv.org/abs/2602.08664

    work page arXiv 2026

  10. [10]

    Theory1(2021), 44,ISSN 2766-1334, Id/No 16

    Darij Grinberg,The Pelletier-Ressayre hidden symmetry for Littlewood- Richardson coefficients, Comb. Theory1(2021), 44,ISSN 2766-1334, Id/No 16

    2021

  11. [11]

    Darij Grinberg,Is the Ford-Fulkerson algorithm a tropical rational func- tion?, MathOverflow, https://mathoverflow.net/q/297015. 12

  12. [12]

    II: Rectangles and triangles, Electron

    Darij Grinberg and Tom Roby,Iterative properties of birational rowmotion. II: Rectangles and triangles, Electron. J. Comb.22(2015), no. 3, research paper p3.40, 49, ISSN 1077-8926

    2015

  13. [13]

    I: Generalities and skeletal posets, Electron

    ,Iterative properties of birational rowmotion. I: Generalities and skeletal posets, Electron. J. Comb.23(2016), no. 1, research paper p1.33, 40, ISSN 1077-8926

    2016

  14. [14]

    Theory3(2023), no

    ,Birational rowmotion on a rectangle over a noncommutative ring, Comb. Theory3(2023), no. 3, 66, ISSN 2766-1334, Id/No 7

    2023

  15. [15]

    Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich,Canonical bases for cluster algebras, J. Am. Math. Soc.31(2018), no. 2, 497–608, ISSN 0894-0347

    2018

  16. [16]

    Modeling and analysis of synchronized systems: a course on max-plus algebra and its applications, Princeton Ser

    Bernd Heidergott, Geert Jan Oldser, and Jacob van der Woude,Max plus at work. Modeling and analysis of synchronized systems: a course on max-plus algebra and its applications, Princeton Ser. Appl. Math., Princeton, NJ: Princeton University Press, 2006, ISBN 0-691-11763-2

    2006

  17. [17]

    Comb.5(2022), no

    Sam Hopkins and Michael Joseph,The birational Lalanne-Kreweras invo- lution, Algebr. Comb.5(2022), no. 2, 227–265, ISSN 2589-5486

    2022

  18. [18]

    Comb.7(2024), no

    Joseph Johnson and Ricky Ini Liu,Birational rowmotion and the octahedron recurrence, Algebr. Comb.7(2024), no. 5, 1453–1477, ISSN 2589-5486

    2024

  19. [19]

    Jeremy Kepner and John Gilbert (eds.),Graph algorithms in the lan- guage of linear algebra, Softw. Environ. Tools, vol. 22, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2011, ISBN 978-0-898719-90-1

    2011

  20. [20]

    Kirillov,Introduction to tropical combinatorics, Physics and combinatorics

    Anatol N. Kirillov,Introduction to tropical combinatorics, Physics and combinatorics. Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, August 21–26, 2000, Singapore: World Scientific, 2001, pp. 82–150, ISBN 981-02-4642-0

    2000

  21. [21]

    Klarner (ed.),Mathematical recreations

    David A. Klarner (ed.),Mathematical recreations. A collection in honor of Martin Gardner, corrected and unabridged republication of the 1981 orig. ed., Mineola, NY: Dover Publications, 1998, ISBN 0-486-40089-1

    1981

  22. [22]

    Kyungyong Lee and Ralf Schiffler,Positivity for cluster algebras, Ann. Math. (2)182(2015), no. 1, 73–125, ISSN 0003-486X

    2015

  23. [23]

    Paul Lezeau, Thomas Walker, Yueqi Cao, Shiv Bhatia, and Anthea Monod, Tropical Expressivity of Neural Networks, Preprint, arXiv:2405.20174 [cs.LG] (2024), 2024, https://arxiv.org/abs/2405.20174. 13

    work page arXiv 2024

  24. [24]

    Diane Maclagan and Bernd Sturmfels,Introduction to tropical geometry, Grad. Stud. Math., vol. 161, Providence, RI: Ameri- can Mathematical Society (AMS), 2015, ISBN 978-0-8218-5198-2, https://web.archive.org/web/20160423075836/https://homepages. warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook20.4.14.pdf

    work page arXiv 2015

  25. [25]

    Comb.2(2019), no

    Gregg Musiker and Tom Roby,Paths to understanding birational rowmo- tion on products of two chains, Algebr. Comb.2(2019), no. 2, 275–304, ISSN 2589-5486

    2019

  26. [26]

    Masatoshi Noumi and Yasuhiko Yamada,Tropical Robinson-Schensted- Knuth correspondence and birational Weyl group action, arXiv:math- ph/0203030v2

    work page arXiv

  27. [27]

    Soichi Okada,Birational rowmotion and Coxeter-motion on minuscule posets, Electron. J. Comb.28(2021), no. 1, research paper p1.17, 30, ISSN 1077-8926

    2021

  28. [28]

    Lior Pachter and Bernd Sturmfels (eds.),Algebraic statistics for com- putational biology, Cambridge: Cambridge University Press, 2005, ISBN 0-521-85700-7

    2005

  29. [29]

    Based on a workshop, Bristol, UK, October 3–7, 1994, Cambridge: Cambridge University Press, 1998, pp

    Jean Eric Pin,Tropical semirings, Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994, Cambridge: Cambridge University Press, 1998, pp. 50–69,ISBN 0-521-55344-X, https://hal.science/hal-00113779v1

    1994

  30. [30]

    Felipe Rincón, Ngoc Mai Tran, and Josephine Yu,Tropical combinatorics, Notices Am. Math. Soc.70(2023), no. 1, 34–43, ISSN 0002-9920

    2023

  31. [31]

    619– 652, ISBN 978-3-319-24296-5; 978-3-319-24298-9

    Tom Roby,Dynamical algebraic combinatorics and the homomesy phe- nomenon, Recent trends in combinatorics, Cham: Springer, 2016, pp. 619– 652, ISBN 978-3-319-24296-5; 978-3-319-24298-9

    2016

  32. [32]

    David Speyer and Bernd Sturmfels,Tropical mathematics, Math. Mag. 82(2009), no. 3, 163–173,ISSN 0025-570X, https://math.berkeley.edu/ ~bernd/mathmag.pdf

    2009

  33. [33]

    Tropical Geometry of Deep Neural Networks

    Liwen Zhang, Gregory Naitzat, and Lek Heng Lim,Tropical Geometry of Deep Neural Networks, Proceedings of the 35th International Conference on Machine Learning (ICML), PMLR (2018), 5824–5832, https://arxiv. org/abs/1805.07091. 14 x z y w b(x+y)w xz b(x+y) z b(x+y)w yz ab z ab y bw(x+y) xy ab x az x+y ayz w(x+y) ab w axz w(x+y) axy w(x+y) x z y w Figure 2: ...

    work page internal anchor Pith review Pith/arXiv arXiv 2018