Dimension of the solution set S(U,γ) for cone-volume vectors

Ascertain whether, for any U ∈ 𝒰(n,m) with unique partition U = S_1 ∪ ⋯ ∪ S_d into irreducible sets and any γ ∈ C_(U) ∩ ℝ^m_{>0}, the semialgebraic dimension of the solution set S(U,γ) = { b ∈ ℝ^m_{≥0} : γ(U,b) = γ } satisfies dim^*(S(U,γ)) = d − 1.

Background

The authors consider the set S(U,γ) of all right-hand sides b producing the same cone-volume vector γ(U,b) = γ, and prove a general lower bound dim^*(S(U,γ)) ≥ d − 1, where d is the number of irreducible components in the canonical partition of U induced by matroid separators. This bound links the multiplicity of representations to the structural decomposability of U.

They conjecture that this lower bound is tight for all U and all strictly positive γ ∈ C_(U), which would imply that the representation is finite (i.e., unique up to finitely many choices) precisely when U is irreducible. This conjecture sharpens understanding of the non-uniqueness in constructing polytopes from cone-volume data and connects it to matroidal structure.