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arxiv: 2511.03342 · v2 · pith:GKPCLST3new · submitted 2025-11-05 · 🧮 math.AG · math.CO

On the piecewise quasipolynomiality of double tropical Welschinger invariants

Vincenzo Reda This is my paper

Pith reviewed 2026-05-18 01:52 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords tropical geometryWelschinger invariantsquasipolynomial functionsh-transverse polygonstoric surfacescombinatorial enumerationreal curves
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The pith

Double tropical Welschinger invariants of toric surfaces from h-transverse polygons are piecewise quasipolynomial.

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

The paper proves that double tropical Welschinger invariants are piecewise quasipolynomial for toric surfaces corresponding to h-transverse polygons, confirming a conjecture in full generality. It extends results previously known only for Hirzebruch surfaces. New combinatorial Welschinger-type numbers are defined for h-transverse polygons, and these are also shown to be piecewise quasipolynomial. This provides a combinatorial approach to counting real curves in these algebraic surfaces.

Core claim

The double tropical Welschinger invariants associated with h-transverse polygons are piecewise quasipolynomial. New combinatorial Welschinger-type numbers for h-transverse polygons are likewise piecewise quasipolynomial.

What carries the argument

The combinatorial definition of the invariants using h-transverse polygons, which enables the piecewise quasipolynomial property to hold uniformly.

If this is right

  • The invariants admit explicit piecewise quasipolynomial expressions that can be used for computation in any such surface.
  • These numbers enumerate real rational curves with given tangency conditions in a tropical setting.
  • The property extends to a new family of combinatorial counts that parallel the tropical invariants.
  • Uniformity across all h-transverse polygons allows for broader applications in enumerative problems.

Where Pith is reading between the lines

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

  • This framework may generalize to other tropical invariants or different classes of polygons.
  • Piecewise quasipolynomiality could lead to efficient algorithms for calculating these numbers in high degree cases.
  • Links to real algebraic geometry might be strengthened by studying how these combinatorial numbers relate to actual curve counts.

Load-bearing premise

The polygons are h-transverse, which is necessary for the combinatorial definitions and the quasipolynomial structure to be valid.

What would settle it

Compute the double tropical Welschinger invariant for a specific h-transverse polygon and check if it fails to be piecewise quasipolynomial as a function of the point positions or degrees.

Figures

Figures reproduced from arXiv: 2511.03342 by Vincenzo Reda.

Figure 1
Figure 1. Figure 1: We can attach two divergence multiplicity vectors to D: 𝜉 = (𝛼, 𝛽,𝛾, 𝛿, 𝛼,˜ ˜𝛽,𝛾,˜ ˜𝛿) = (1, 0, 0, 0, 1, 0, 0001, 01) 𝜉 ′ = (𝛼 ′ , 𝛽′ ,𝛾′ , 𝛿′ , 𝛼˜ ′ , ˜𝛽 ′ ,𝛾˜ ′ , ˜𝛿 ′ ) = (1, 0, 0, 0, 1, 02, 0001, 0). The divergence sequence associated to 𝜉 is (x𝜉 ; y𝜉 ; z𝜉 ; w𝜉 ) = (1, −1; 0; 4; 2), while the divergence sequence associated to 𝜉 ′ is (x𝜉 ′; y𝜉 ′; z𝜉 ′; w𝜉 ′) = (1, −1; 2, 2; 4; 0). Let us compute the mul… view at source ↗
Figure 2
Figure 2. Figure 2: 𝑥1 > 0 𝑥1 + 𝑘 > 0 𝑦1 > 0 𝑦1 + 𝑘 > 0 𝑦2 + 𝑘 > 0 𝑦2 > 0 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hyperplane arrangement in Λ chamber structure is fixed under permutation of the 𝑛2 y variables. What remains to prove is that ∑︁ (𝑟,𝑙) ∑︁ 𝜎 ∈𝑆𝑛2 𝐺𝐺,c,𝑟−𝑙(x, 𝜎(y)) is piecewise quasipolynomial. In order to do so, note that, in general, 𝑆𝑟−𝑙 is not the same hyperplane arrangement as 𝑆 𝑟˜−˜ 𝑙 for 𝑟 ≠ 𝑟˜ and 𝑙 ≠ ˜ 𝑙 (see [HR24]). Therefore, when we sum over all the pairs (𝑟,𝑙), the resulting map will be piecew… view at source ↗
Figure 4
Figure 4. Figure 4: that originate the permutations 𝑟1 = (𝑘, 0), 𝑟2 = (0, 𝑘) and 𝑙 = (0, 0). Therefore, black vertices can have divergences given by 𝑟1 − 𝑙 = (𝑘, 0), 𝑟2 − 𝑙 = (0, 𝑘). We compute the maps 𝐺 1,2 (d 𝑟 ;d 𝑙 ),c,1 (𝑥1, 𝑦1, 𝑦2) and 𝐺 1,2 (d 𝑟 ;d 𝑙 ),1 (𝑥1, 𝑦1, 𝑦2, 𝑘) in only one chamber of the hyperplane arrangement. The hyperplane arrangement in Λ = {(𝑥1, 𝑦1, 𝑦2) ∈ Z 3 |𝑥1 + 𝑦1 + 𝑦2 + 𝑘 = 0} is given by hyperplanes… view at source ↗
Figure 5
Figure 5. Figure 5: while I (D2,𝑚2, 1) = {𝑚2 (2),𝑚2 (3)}. If 𝜀˜ is the sequence given by (0, 0, −1), one can easily see that ˜𝛽 ′ = ˜𝛽 − 2𝜀˜ and ˜𝛿 ′ = ˜𝛿 +𝜀˜. This operation tells us that we eliminate an 𝑠-pair from the imaginary part. Remark 4.2. Example 2.9 shows that two divergence sequences in the same equivalence class cannot be attached to the same 𝑠-real floor diagram. We encode the combinatorial Welschinger-type numb… view at source ↗
Figure 6
Figure 6. Figure 6: Hyperplane arrangement in Λ˜ . The chambers in red are the chambers in which we compute the map 𝐺 1,2 (d 𝑟 ;d 𝑙 ),c,0,1 (𝑥1, 𝑦1, 𝑦2, 0). We denote by C1 the chamber given by the inequalities −𝑘 < 𝑥1 < 0, 𝑦1 < −𝑘 and −𝑘 < 𝑦2 < 0 and C2 the chamber given by the inequalities 𝑥1 < −𝑘 and 𝑦1, 𝑦2 > 0. The choice of the chambers is not random: in the chamber C1, 𝑦1 = 𝑦2 cannot happen, therefore the floor diagrams… view at source ↗
Figure 7
Figure 7. Figure 7: The floor diagram 𝐶1 with two different weightings. so they are 1-real floor diagram having multiplicity −𝑦1 and −𝑦2 respectively, as long as 𝑥1, 𝑦1, 𝑦2, 𝑘 are odd. We distinguish, as in Section 3.3, two cases. If 𝑘 ≥ 0 is even, 𝑥1 is even as long as 𝑦1 and 𝑦2 are odd. Therefore, in this case 𝐺 1,2 (d 𝑟 ;d 𝑙 ),c,0,1 (𝑥1, 𝑦1, 𝑦2, 0) = 0. If 𝑘 ≥ 0 is odd, the graphs contributing non-zero to 𝐺 1,2 (d 𝑟 ;d 𝑙 )… view at source ↗
Figure 8
Figure 8. Figure 8 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

Ardila and Brugall\'e conjectured that double tropical Welschinger invariants of Hirzebruch surfaces are piecewise quasipolynomial. In this work, we prove the conjecture holds in full generality, i.e. for toric surfaces corresponding to h-transverse polygons. Furthermore, we define new combinatorial Welschinger-type numbers for h-transverse polygons and show that they are likewise piecewise quasipolynomial.

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

1 major / 2 minor

Summary. The manuscript proves the Ardila-Brugallé conjecture that double tropical Welschinger invariants of Hirzebruch surfaces are piecewise quasipolynomial, extending the result to toric surfaces corresponding to h-transverse polygons. It introduces new combinatorial Welschinger-type numbers for h-transverse polygons and establishes that these numbers are likewise piecewise quasipolynomial, using explicit combinatorial definitions, recursive relations, and case-by-case verifications internal to the h-transverse class.

Significance. If the central claims hold, the work provides a combinatorial framework that generalizes known quasipolynomiality results from Hirzebruch surfaces to a wider class of toric surfaces. The explicit definitions and internal verifications strengthen the combinatorial approach to tropical enumerative invariants and may enable further explicit computations and extensions in the field.

major comments (1)
  1. The extension from Hirzebruch surfaces to general h-transverse polygons is handled by direct specialization of the combinatorial data, but the manuscript should explicitly verify that the chamber decomposition and quasipolynomial degree remain unchanged under this specialization (see the argument following the definition of the new numbers).
minor comments (2)
  1. Clarify the notation for the recursive relations in the combinatorial definitions to ensure they are self-contained without reference to prior tropical Welschinger literature.
  2. Add a brief remark on how the h-transverse condition is used in the base cases of the verification to improve readability for readers unfamiliar with the polygon class.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for minor revision. We respond to the major comment below.

read point-by-point responses

  1. Referee: The extension from Hirzebruch surfaces to general h-transverse polygons is handled by direct specialization of the combinatorial data, but the manuscript should explicitly verify that the chamber decomposition and quasipolynomial degree remain unchanged under this specialization (see the argument following the definition of the new numbers).

    Authors: We thank the referee for this observation. The proofs in the manuscript are carried out uniformly for arbitrary h-transverse polygons, so that the Hirzebruch case is recovered by direct specialization of the polygon data. The chamber decomposition is determined by the hyperplanes in the parameter space where the recursive relations for the new combinatorial numbers change; these hyperplanes specialize without modification when the polygon is restricted to the Hirzebruch subclass, and the quasipolynomial degree (fixed by the number of marked points and the surface dimension) is likewise invariant. To make the preservation explicit as requested, we will insert a short clarifying paragraph immediately after the definition of the new numbers, confirming that both the chamber decomposition and the degree remain unchanged under this specialization. The revision will be included in the next version of the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit combinatorial definitions

full rationale

The paper proves the Ardila-Brugallé conjecture in full generality for h-transverse polygons by introducing new combinatorial Welschinger-type numbers and verifying their piecewise quasipolynomiality through explicit combinatorial definitions, recursive relations, and direct case-by-case verifications internal to the h-transverse class. The extension from Hirzebruch surfaces occurs via specialization of the same data without reducing any central claim to a fitted input, self-referential equation, or load-bearing self-citation chain. All steps remain independent of the target result and are externally falsifiable via the stated combinatorial constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the combinatorial definitions of double tropical Welschinger invariants and h-transverse polygons from prior literature; no free parameters, new axioms, or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Standard definitions of tropical Welschinger invariants and h-transverse polygons from Ardila-Brugallé and related tropical geometry literature
    The proof extends these existing combinatorial objects to the general case.

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Floor decompositions of tropical curves : the planar case

    [AB13] F. Ardila and F. Block. “Universal polynomials for Severi degrees of toric surfaces”. In: Advances in Mathematics237 (2013), pp. 165–193. [AB17] F. Ardila and E. Brugallé. “The double Gromov–Witten invariants of Hirzebruch surfaces are piecewise polynomial”. In:International Mathematics Research Notices2017.2 (2017), pp. 614–641. [ABD11] A. Arroyo,...

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  2. [2]

    Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

    [Wel05] J.-Y. Welschinger. “Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry”. In:Inventiones mathematicae162.1 (2005), pp. 195–234. V. Reda: School of Mathematics, 17 Westland Row, Trinity College Dublin, Dublin 2, Ireland Email address:redav@tcd.ie

    work page 2005