Quadratically Enriched Tropical Curve Invariants

Definitions and mathematical statement

Let $a,b\in{\mathbb{Z}}_{\geq 0}$. The supplied record names two families of quadratically enriched tropical curve invariants, denoted by $N^{\mathbb A^1}_{u,v}$ and $N^{\mathbb A^1}_{\triangle,u,v}$. Its fragmentary original text refers to a corresponding toric surface and an ample divisor, but does not specify these geometric data or give definitions of the invariants. In particular, the base field, coefficient ring, admissible indices, curve classes, incidence conditions, and enrichment conventions are not supplied.

The stated conjecture is that, for every pair of integers $a,b\geq 0$, \[ N^{\mathbb A^1}_{a,a+b} = \sum_{k\geq 0}\binom{b+2k}{k} N^{\mathbb A^1}_{\triangle,a-k,b+2k}. \] Here the sum is indexed by nonnegative integers $k$, and \[ \binom{b+2k}{k}=\frac{(b+2k)!}{k!\,(b+k)!}. \] The task is to establish or disprove this identity with the intended geometric definitions and hypotheses.

Unresolved qualifications. For $k>a$, the first numerical index $a-k$ on the right is negative. The supplied record gives neither a definition at such indices nor a vanishing convention. Thus the displayed sum is retained exactly as supplied; replacing it by a sum over $0\leq k\leq a$ would require an additional justification or convention. The record also does not specify a finiteness or convergence interpretation for the sum, or the algebraic setting in which the integer coefficients act. These omissions prevent a fully specified mathematical statement without consulting the underlying definitions.

The extracted original text additionally says that an analogous conjecture for tropical refined invariants was formulated in its reference [5] and proved by Bousseau in its reference [2]. This is a supplied background claim about an analogous relation, not a supplied proof of the quadratically enriched identity displayed above.

Short English statement

The conjecture expresses a quadratically enriched tropical curve invariant as a binomially weighted sum of another family of such invariants, for every $a,b\geq 0$. The geometric definitions, coefficient setting, and treatment of negative indices are missing from the supplied record and require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005534$, title Quadratically Enriched Tropical Curve Invariants, source problem code catalogue entry. The record assigns the problem to 2023 and names no proposer.
• Tropical Methods in Geometry (2023), Oberwolfach Reports, pp. 1263--1322, DOI 10.4171/owr/2023/23: https://doi.org/10.4171/owr/2023/23. The supplied extraction identifies the displayed target as Conjecture 1.
• Supplied literature source labelled "OWR 2023 23": https://ems.press/content/serial-article-files/47020.
• Supplied literature source labelled "Jaramillo Puentes Markwig Pauli Röhrle": https://arxiv.org/abs/2502.02569. No title or specific supporting passage is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that later correspondence and algorithmic results do not establish this exact identity. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports a direct check of the official report. These are supplied provenance claims; no independent source verification is asserted here.
• The original extraction refers to an analogous refined conjecture in its reference [5] and a proof by Bousseau in its reference [2], but supplies neither bibliography entry. Its acknowledgment states partial support by ANR project ENUM-GEOM NR-18-CE40-0009-02.