Coefficient Criterion for Redundant Tropical String Inequalities

Definitions and mathematical statement

Write the potential in the chart denoted by $T_i^\vee$ as a Laurent polynomial \[ f=W|_{T_i^\vee}=\sum_{a\in A}c_a x^a, \qquad x^a=\prod_{k=1}^{N}x_k^{a_k}, \] where $A\subset{\mathbb{Z}}^N$ is finite and $c_a\in{\mathbb{C}}\setminus\{0\}$. The supplied excerpt also uses a chart $T_\Sigma^\vee$ and asserts that all exponents of $W|_{T_\Sigma^\vee}$ are nonpositive. It does not define the indexing data $i,\Sigma$, their relationship, the potential $W$, or the admissible family of charts.

Under coefficient-free minimum tropicalization, the Laurent polynomial determines the piecewise linear function \[ f^{\mathrm{trop}}(u)=\min_{a\in A}\langle a,u\rangle, \qquad u\in{\mathbb{R}}^N. \] This specifies the convention in which nonzero coefficients contribute no tropical constant. The excerpt describes multiplication becoming addition and addition becoming minimum, but does not supply a coefficient valuation or a complete definition of the resulting inequality system. In particular, its truncated formula does not reliably specify the tropical domain.

To express redundancy without supplying missing inequality conventions, let $D$ denote the intended ambient domain and let $I_a(u)$ denote the inequality associated with the term $c_ax^a$. Set \[ P=\{u\in D:I_b(u)\text{ holds for every }b\in A\}. \] The inequality $I_a$ is redundant precisely when deleting it leaves $P$ unchanged, equivalently when \[ \forall u\in D,\qquad \left(\bigwedge_{b\in A\setminus\{a\}}I_b(u)\right) \Longrightarrow I_a(u). \] Any additional fixed constraints must be incorporated into $D$.

Recorded conjecture, with unresolved qualifications. The clean statement and the extracted Conjecture 4 ask whether, for every admissible potential and chart and every term $c_ax^a$, \[ I_a\text{ is redundant}\quad\Longleftrightarrow\quad c_a\leq 1. \] This is a schematic formulation: the admissible data, the ambient domain, the individual inequalities, and an ordered coefficient domain remain unspecified. The displayed comparison requires real coefficients or another explicitly ordered coefficient domain; membership in a complex Laurent polynomial ring alone does not define it.

Conflicting supplied formulation. The background and literature assessment instead describe the criterion \[ I_a\text{ is redundant}\quad\Longleftrightarrow\quad c_a>1. \] For real coefficients, these are incompatible proposed characterizations. The supplied evidence does not resolve which comparison is intended, so neither is selected as a corrected conjecture.

Reported background qualification. The excerpt labels as Theorem 3, attributed there to reference [3], the assertion that if every variable exponent in $W|_{T_i^\vee}$ belongs to $\{-1,0\}$, then the inequalities of $C_{\Sigma_i}$, and hence those of $S_i$, are nonredundant. The objects $C_{\Sigma_i}$, $S_i$, the full hypotheses, and the identity of reference [3] are not supplied. This background assertion is retained as reported context, not as an independently verified theorem or a resolution of the conflicting coefficient criteria.

Short English statement

The question asks whether the coefficient of a monomial in a chart expression of the potential exactly determines whether its tropical inequality can be deleted without changing the feasible set. The recorded conjecture uses coefficient at most $1$, while the supplied literature assessment uses coefficient greater than $1$. The intended comparison, coefficient assumptions, chart data, and inequality system require clarification before a definitive mathematical statement can be accepted.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004896$, source problem code catalogue entry, titled "Coefficient Criterion for Redundant Tropical String Inequalities." The record gives proposed year 2021 and does not identify a proposer.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2022), Oberwolfach Reports, pp. 2827--2891, DOI 10.4171/owr/2021/52. https://doi.org/10.4171/owr/2021/52. The supplied excerpt labels the target Conjecture 4 and refers to a preceding Theorem 3 with an unresolved reference [3].
• Supplied literature source: Koshevoy--Schumann, Journal of Algebraic Combinatorics. https://doi.org/10.1007/s10801-022-01144-z. No article title, precise result location, or full bibliography is supplied.
• Supplied literature source: "Oberwolfach Report 52, string cones." https://doi.org/10.4171/owr/2021/52.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the coefficient-greater-than-one criterion remains conjectural, with examples and sufficient criteria but no general proof or counterexample in the primary paper. This is supplied dated curation, not independently verified literature assessment; its coefficient comparison conflicts with the recorded conjecture.
• The registry labels the statement corrected_verified and reports that the official report was checked directly. That verification claim is supplied provenance and has not been independently verified in this draft.