Convergence and positivity of the general TU capacity lower bound

Determine whether the lower bound in Theorem \Cref{thm: initial capacity bound for tu matrices} converges to a nonzero limit as the truncation vector $\bm{k}$ tends coordinatewise to infinity for general totally unimodular matrices with finite fibers.

Background

For truncated generating polynomials, the paper establishes capacity lower bounds for lattice-point fibers of totally unimodular systems. In special cases involving nonnegative partial row sums, it proves that the capacities converge as the truncation bounds tend to infinity, allowing the truncated estimates to yield bounds for the untruncated fibers.

For general totally unimodular matrices, the paper uses a different argument because it does not establish either convergence of the lower bound from Theorem \Cref{thm: initial capacity bound for tu matrices} or positivity of any such limit.

References

We remark that we took a different approach in \Cref{subsection: general capacity bounds via certificate of boundedness} when working with general TU matrices. This is because it is not clear whether the lower bound of \Cref{thm: initial capacity bound for tu matrices} converges as $\bm{k} \to \infty$, or if it converges, whether it converges to a non-zero number.

— Log-concavity and Approximate Counting for Totally Unimodular Polytopes  (2609.39917 - Leake et al., 30 Sep 2026) in Section 6, subsection 'Special cases: non-negative partial row sums'