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arxiv: 2603.25953 · v2 · submitted 2026-03-26 · 🧮 math.AG · math.AC

Geometric classification of primes modulo a (bend) congruence

Netanel Friedenberg , Kalina Mincheva This is my paper

Pith reviewed 2026-05-14 23:50 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords tropical geometryprime congruencestoric semiringsbend congruenceNullstellensatztropical idealscancellative semiringstropical function semiring
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The pith

Prime congruences containing a given congruence on a toric semiring admit strong geometric characterizations.

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

The paper establishes strong characterizations of prime congruences that contain a fixed congruence on a toric semiring. These characterizations produce an analogue of the strong Nullstellensatz that applies even when the congruence has only a finite tropical basis, covering cases such as the bend congruence of a tropicalized ideal. The results also imply that certain quotient semirings are cancellative and coincide with the tropical function semiring on the corresponding variety. Finally, they yield a description of the closure of a polyhedron inside a tropical toric variety regardless of fan compatibility.

Core claim

We give strong characterizations of prime congruences containing a given congruence on a toric semiring. This yields an analogue of the strong Nullstellensatz for congruences with finite tropical basis, including the bend congruence of a tropicalized ideal. When I is the ideal of an affine variety not contained in the coordinate hyperplanes, the quotient by the radical of the bend congruence is cancellative and equals the tropical function semiring on the tropical variety. As a side consequence, the closure of any polyhedron inside a tropical toric variety can be described even when the polyhedron is incompatible with the defining fan.

What carries the argument

Strong characterizations of prime congruences containing a given congruence on a toric semiring, which classify the primes geometrically and support the Nullstellensatz analogue.

If this is right

  • An analogue of the strong Nullstellensatz holds for any congruence possessing a finite tropical basis.
  • For the ideal I of an affine variety outside the coordinate hyperplanes, the semiring obtained by quotienting by the radical of the bend congruence is cancellative.
  • The same quotient semiring coincides with the tropical function semiring on the tropical variety trop V(I).
  • The closure of any polyhedron inside a tropical toric variety is describable even without fan compatibility.

Where Pith is reading between the lines

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

  • The classification may extend the bridge between algebraic non-embedded tropicalization and bend-congruence methods to additional classes of ideals.
  • Cancellativity opens a route to computing integral closures of these semirings in a follow-up setting.
  • The polyhedron-closure description could apply directly to polyhedral fans that arise in non-toric ambient spaces.

Load-bearing premise

The given congruence admits a finite tropical basis, or the affine variety is not contained in the coordinate hyperplanes.

What would settle it

A concrete prime congruence containing the bend congruence of a tropical ideal whose radical does not correspond to a point of the tropical variety, or whose quotient fails to be cancellative.

read the original abstract

In this paper we continue the program to develop the algebraic foundations of tropical (algebraic) geometry. We give strong characterizations of prime congruences containing a given congruence on a toric semiring. We give four applications of this result. (1) We prove an analogue of the strong Nullstellensatz for congruences with finite tropical basis. This extends the existing result of Jo\'o-Mincheva to cases, such as the bend congruence of a tropical(ized) ideal, where the congruence is not finitely generated. (2) We show that, if $I$ is the ideal of an affine variety not contained in the coordinate hyperplanes, then $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is cancellative. This result has applications to the integral closure (as per Tolliver) of $\mathbb{T}[x_1, \dots, x_n]/\operatorname{Bend}(\operatorname{trop} I)$ which we explore in a forthcoming paper. (3) We show that $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is the tropical function semiring on $\operatorname{trop} V(I)$, which creates a bridge between the algebraic approach to non-embedded tropicalization in the work of J. Song and the bend congruence approach of Giansiracusa-Giansiracusa and Maclagan-Rinc\'on. (4) As a consequence of one of our lemmas, we describe the closure of a polyhedron in a tropical toric variety even when the polyhedron is not compatible with the fan defining the tropical toric variety.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript gives strong characterizations of prime congruences containing a given congruence on a toric semiring. It applies the result to four settings: an analogue of the strong Nullstellensatz for congruences possessing a finite tropical basis (including bend congruences of tropicalized ideals), cancellativity of T[x1,...,xn]/sqrt(Bend(trop I)) when the affine variety is not contained in the coordinate hyperplanes, identification of the same quotient as the tropical function semiring on trop V(I), and a description of the closure of a polyhedron inside a tropical toric variety even when the polyhedron is not compatible with the defining fan.

Significance. If the characterizations are correct, the work strengthens the algebraic foundations of tropical geometry by extending Nullstellensatz-type theorems beyond finitely generated congruences and by linking the algebraic approach of Song with the bend-congruence framework of Giansiracusa-Giansiracusa and Maclagan-Rincón. The cancellativity statement supplies a concrete tool for studying integral closures, while the polyhedral-closure lemma has independent geometric utility.

minor comments (3)
  1. The abstract refers to 'Joó-Mincheva' and 'Giansiracusa-Giansiracusa'; the introduction should supply the precise citations (including arXiv numbers or journal details) for these prior results.
  2. Notation such as Bend(trop I) and sqrt(Bend(trop I)) is introduced in the abstract; a short preliminary section or table collecting all semiring and congruence notation would improve readability for readers outside the immediate subfield.
  3. Application (4) on polyhedral closure is stated without an accompanying figure or low-dimensional example; adding one would make the statement more concrete.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, which correctly identifies the core characterizations of prime congruences containing a given congruence on toric semirings and the four applications. We appreciate the recommendation for minor revision and the recognition that the work strengthens algebraic foundations in tropical geometry by extending Nullstellensatz-type results and bridging algebraic and bend-congruence approaches. No specific major comments were raised in the report, so we will incorporate minor revisions as appropriate in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes a classification of prime congruences containing a given congruence on toric semirings via direct algebraic arguments on the structure of congruences and tropical bases. The strong Nullstellensatz analogue is derived from this classification once a finite tropical basis is assumed, extending (but not reducing to) the Joó-Mincheva result. The cancellativity and function-semiring statements follow from the classification under the explicit non-containment hypothesis. No self-definitional equations, fitted inputs renamed as predictions, load-bearing self-citations, or ansatz smuggling appear; all steps rest on internal definitions and lemmas rather than circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard axioms of commutative semirings and toric varieties from the tropical geometry literature, with no new free parameters or invented entities introduced.

axioms (2)
  • standard math Toric semirings are commutative semirings equipped with a toric structure compatible with the tropical operations.
    Invoked throughout as background for the congruence theory.
  • domain assumption The bend congruence is defined via the Giansiracusa-Giansiracusa construction on tropical polynomials.
    Used as the given congruence in the applications.

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    Akian, A

    M. Akian, A. Béreau, S. GaubertThe Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Sys- tems, arxiv preprint arXiv:2312.05859

    work page arXiv

  2. [2]

    Aliprantis, K

    C. Aliprantis, K. Border,Infinite Dimensional Analysis: A Hitchhiker’s Guide, Springer Berlin, Heidelberg, DOI https://doi.org/10.1007/978-3-662-03961-8 (2013)

    work page doi:10.1007/978-3-662-03961-8 2013

  3. [3]

    Amini, S

    O. Amini, S. Kawaguchi, J. SongTropical function fields, finite generation, and faithful tropicalizationarXiv preprint arXiv:2503.20611 (2025)

    work page arXiv 2025

  4. [4]

    Baker, J

    M. Baker, J. Rabinoff,The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves, Int. Math. Res. Not. (16) 7436–7472, (2015)

    work page 2015

  5. [5]

    Bertram and R

    A. Bertram and R. Easton,The Tropical Nullstellensatz for Congruences, Advances in Mathematics 308 (2017) 36-82

    work page 2017

  6. [6]

    D. Cox, J. Little, H. Schenck,Toric Varieties, volume 124 of Graduate Studies in Mathematics. Amer. Math. Soc., Providence, RI, (2011)

    work page 2011

  7. [7]

    Hrushovski, F

    A.Ducros, E. Hrushovski, F. Loeser, J. Ye.Tropical functions on a skeleton, Journal de l’ École Polytechnique, 11:613–654, (2024)

    work page 2024

  8. [8]

    Ewald, M

    G. Ewald, M. Ishida,Completion of real fans and Zariski-Riemann spaces, Tohoku Math. J. (2) 58 (2) 189 - 218, https://doi.org/10.2748/tmj/1156256400, (2006)

    work page doi:10.2748/tmj/1156256400 2006

  9. [9]

    A. Fink, J. Giansiracusa, N. Giansiracusa, J. MundingerProjective hypersurfaces in tropical scheme theory I: the Macaulay idealarxiv preprint arXiv:2405.16338 (2024)

    work page arXiv 2024

  10. [10]

    Friedenberg, K

    N. Friedenberg, K. Mincheva,Tropical Adic Spaces I: The continuous spectrum of a topological semiring, Res. Math. Sci. 11, 55 (2024). DOI: 10.1007/s40687-024-00467-6

    work page doi:10.1007/s40687-024-00467-6 2024

  11. [11]

    Friedenberg, K

    N. Friedenberg, K. Mincheva,Geometric interpretation of valuated term (pre)orders, arXiv preprint, arXiv:2306.05538 (2023) GEOMETRIC CLASSIFICATION OF PRIMES MODULO A (BEND) CONGRUENCE 38

    work page arXiv 2023

  12. [12]

    Friedenberg, K

    N. Friedenberg, K. Mincheva,Integral closure for semirings, forthcoming

    work page

  13. [13]

    Fulton,Introduction to toric varieties, Annals of Mathematics Studies, vol

    W. Fulton,Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Prince-ton, NJ, 1993, The William H. Roever Lectures in Geometry

    work page 1993

  14. [14]

    Giansiracusa, N

    J. Giansiracusa, N. Giansiracusa,Equations of tropical varieties, Duke Math. J. Volume 165, Number 18 (2016), 3379-3433

    work page 2016

  15. [15]

    Grigoriev and V

    D. Grigoriev and V . Podolskii,Tropical effective primary and dual Nullstellensätze, Discrete & Computational Geometry, Volume 59, Issue 3, Pages 507 – 552, https://doi.org/10.1007/s00454-018-9966-3 (2018)

    work page doi:10.1007/s00454-018-9966-3 2018

  16. [16]

    Gubler,A guide to tropicalizations, Algebraic and Combinatorial Aspects of Tropical Geometry, Contem- porary Mathematics, vol 589, DOI: https://doi.org/10.1090/conm/589 (2013)

    W. Gubler,A guide to tropicalizations, Algebraic and Combinatorial Aspects of Tropical Geometry, Contem- porary Mathematics, vol 589, DOI: https://doi.org/10.1090/conm/589 (2013)

    work page doi:10.1090/conm/589 2013

  17. [17]

    Gubler, J

    W. Gubler, J. Rabinoff, A. Werner,Skeletons and tropicalizations, Advances in Mathematics, Volume 294, 150– 215 (2016)

    work page 2016

  18. [18]

    ItoNon-finite generatedness of the congruences defined by tropical varieties, arxiv preprint ArXiv: 2512.21565 (2026)

    T. ItoNon-finite generatedness of the congruences defined by tropical varieties, arxiv preprint ArXiv: 2512.21565 (2026)

    work page arXiv 2026

  19. [19]

    Izhakian and L

    Z. Izhakian and L. Rowen,Congruences and coordinate semirings of tropical varieties, Bulletin des Sciences Mathématiques Volume 140, Issue 3 (2016), 231-259

    work page 2016

  20. [20]

    D. Joó, K. Mincheva,Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials, Sel. Math. New Ser. (2017), doi:10.1007/s00029-017-0322-x

    work page doi:10.1007/s00029-017-0322-x 2017

  21. [21]

    D. Joó, K. Mincheva,On the dimension of the polynomial and the Laurent polynomial semiring, J. of Algebra, vol. 507, pp. 103–119 (2018)

    work page 2018

  22. [22]

    D. Joó, K. Mincheva,Varieties of prime tropical ideals and the dimension of the coordinate semiring arXiv:2501.18053 (2025)

    work page arXiv 2025

  23. [23]

    Maclagan, F

    D. Maclagan, F. Rincón,Tropical schemes, tropical cycles, and valuated matroids, J. Eur. Math. Soc. 22 (2020), no. 3, pp. 777–796

    work page 2020

  24. [24]

    Maclagan, F

    D. Maclagan, F. Rincón,Tropical ideals, Compositio Mathematica, Volume 154, Issue 3, 640 – 670, (2018), DOI: S0010437X17008004

    work page 2018

  25. [25]

    Maclagan, F

    D. Maclagan, F. Rincón,Varieties of tropical ideals are balanced, Advances in Mathematics, Volume 410, Part A, (2022)

    work page 2022

  26. [26]

    Maclagan, B

    D. Maclagan, B. Sturmfels,Introduction to Tropical GeometryGraduate Studies in Mathematics 161, American Mathematical Society, (2015)

    work page 2015

  27. [27]

    Mincheva,Prime congruences and tropical geometry, PhD Thesis, Johns Hopkins University (2016)

    K. Mincheva,Prime congruences and tropical geometry, PhD Thesis, Johns Hopkins University (2016)

    work page 2016

  28. [28]

    OdaConvex Bodies and Algebraic GeometryBerlin [u.a.]: Springer, (1988)

    T. OdaConvex Bodies and Algebraic GeometryBerlin [u.a.]: Springer, (1988). <http://eudml.org/doc/203658>

    work page 1988

  29. [29]

    PayneAnalytification is the limit of all tropicalizationsMath

    S. PayneAnalytification is the limit of all tropicalizationsMath. Res. Lett. 19 (3), 543–556, (2009)

    work page 2009

  30. [30]

    Rabinoff,Tropical analytic geometry, Newton polygons, and tropical intersectionsAdvances in Mathematics vol

    J. Rabinoff,Tropical analytic geometry, Newton polygons, and tropical intersectionsAdvances in Mathematics vol. 229, issue 6, p. 3192–3255, (2012)

    work page 2012

  31. [31]

    Rohrer,Completions of fans, J

    F. Rohrer,Completions of fans, J. Geom. 100, 147 (2011). https://doi.org/10.1007/s00022-011-0073-3

    work page doi:10.1007/s00022-011-0073-3 2011

  32. [32]

    J. Song, Y. Nakajima,A geometric interpretation of Krull dimensions of T-algebrasarxiv preprint ArXiv:2408.02366 (2024)

    work page arXiv 2024

  33. [33]

    Song,Finitely generated congruences on tropical rational function semifieldsarxiv preprint ArXiv:2405.14087 (2024)

    J. Song,Finitely generated congruences on tropical rational function semifieldsarxiv preprint ArXiv:2405.14087 (2024)

    work page arXiv 2024

  34. [34]

    Congruences on tropical rational function semifields and tropical curves.arXiv preprint arXiv:2305.04204,

    J. Song,Congruences on tropical rational function semifields and tropical curvesarxiv preprint ArXiv:2305.04204 (2023)

    work page arXiv 2023

  35. [35]

    Song,T-algebra homomorphisms between rational function semifields of tropical curvesarxiv preprint ArXiv:2304.03508 (2023)

    J. Song,T-algebra homomorphisms between rational function semifields of tropical curvesarxiv preprint ArXiv:2304.03508 (2023)

    work page arXiv 2023

  36. [36]

    Extension of valuations in characteristic one

    J. Tolliver,Extension of valuations in characteristic onearxiv preprint ArXiv:1605.06425

    work page internal anchor Pith review Pith/arXiv arXiv

  37. [37]

    N. Tran, J. Wang,Minimal Representations of Tropical Rational Functionsarxiv preprint ArXiv:2205.05647 DEPARTMENT OFMATHEMATICS, TULANEUNIVERSITY, NEWORLEANS, LA 70118, USA Email address:nfriedenberg@tulane.edu DEPARTMENT OFMATHEMATICS, TULANEUNIVERSITY, NEWORLEANS, LA 70118, USA Email address:kmincheva@tulane.edu

    work page arXiv