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PreprintCC BY 4.0arXiv

A note on bounded ratios

  • Lorenzo Baldi
  • Mario Kummer
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Abstract

We prove that the set of bounded ratios BR(X) on a semialgebraic set X⊂Rn>0 is the convex cone of linear forms that are nonnegative on the tropicalization trop(X). In particular, it is a rational polyhedral convex cone. For X the set of Lorentzian polynomials with fixed M-convex support, it is the dual to the set of M-convex functions. We record an explicit counterexample to a conjecture of Huang--Huh--Soskin--Wang on the bounded ratios on Lorentzian polynomials. The bounded ratio in the counterexample corresponds to the non-hypermetric clique-web facet CW17(1,1,1,1,1,−1,−1) of the cut cone on seven vertices.

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Cite

BibTeXRIS
BibTeX
@misc{baldi2026note,
  title = {{A note on bounded ratios}},
  author = {Lorenzo Baldi and Mario Kummer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03934},
  eprint = {2609.03934},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03934},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A note on bounded ratios
AU  - Lorenzo Baldi
AU  - Mario Kummer
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03934
UR  - https://arxiv.org/abs/2609.03934
AB  - We prove that the set of bounded ratios BR(X) on a semialgebraic set X⊂R^n_>0 is the convex cone of linear forms that are nonnegative on the tropicalization trop(X). In particular, it is a rational polyhedral convex cone. For X the set of Lorentzian polynomials with fixed M-convex support, it is the dual to the set of M-convex functions. We record an explicit counterexample to a conjecture of Huang--Huh--Soskin--Wang on the bounded ratios on Lorentzian polynomials. The bounded ratio in the counterexample corresponds to the non-hypermetric clique-web facet CW^1_7(1,1,1,1,1,−1,−1) of the cut cone on seven vertices.
ER  -