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Research Article · Preprint Mathematics

Power Concavity of Low-Dimensional Unimodular Fiber Counts

Scott Schubr

  • Mathematics
  • Combinatorics and Graph Theory
Primary subject
Mathematics
Subject subfields
Combinatorics and Graph Theory
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

We prove weighted power-concavity inequalities for lattice-point counts of bounded unimodular fibers. In central affine dimension at most two, the square root of the corrected count N − 1/4 satisfies weighted Jensen; the correction 1/4 is best possible. If every endpoint count is at least m ≥ 1, the power (2m − 1)/(4m − 1) is valid. The best universal power obeys A(m) = 1/2 − 1/(8m) + O(m^{−3/2}), with explicit finite-threshold bounds. The obstruction uses a fixed six-edge network. We also prove that 4/13 is the optimal concavity power of the classical continuous piecewise polynomial extension of the type-A₃ Kostant partition function. For actual integral demands, cube-root concavity already fails on this four-vertex graph, including a midpoint triple with three-dimensional fibers. An explicit lattice-only upper bound complements the 4/13 lower bound, without determining the optimal lattice exponent. The planar argument combines unimodular elimination, Pick’s theorem and classical Brunn–Minkowski concavity; the A₃ argument uses the classical chamber formulas and a Hessian comparison. These results give restricted global Jensen inequalities; the unrestricted unimodular-fiber question remains open.

Keywords unimodular fibers · integer flows · power concavity · lattice-point counts · Kostant partition function · Brunn–Minkowski inequality

Scope of the work

Weighted Jensen inequalities apply under the stated bounded unimodular-fiber and central-dimension hypotheses. The continuous type-A₃ exponent 4/13 is optimal; the optimal lattice-only exponent is bounded but not determined. Unrestricted higher-dimensional unimodular-fiber Jensen remains open.

Reproduction notes

The six-member package contains the 9-page paper, complete source and bibliography, an optional exact lattice-witness checker, reader instructions, citation metadata and checksums. With Python 3.9 or later, run verify_counterexample.py in normal and optimized modes. Build main.tex with Tectonic or two pdflatex runs. The checker reconstructs finite lattice witnesses; the concavity results rely on the analytic proofs.

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Author disclosures

AI-assisted tools were used in developing and internally checking the arguments and preparing the manuscript. These checks are not external refereeing or formal proof-assistant verification. Classical convex geometry and chamber enumeration are explicitly credited; no new continuous Wills inequality or chamber-count formula is claimed. External novelty and worldwide priority remain provisional.

References and related publications

  1. J. Leake and M. Mohammadi Yekta, Log-concavity and Approximate Counting for Totally Unimodular Polytopes, arXiv:2609.39917v1 (2026). arxiv.org/abs/2609.39917.
  2. A. Khare, J. P. Matherne and A. St. Dizier, Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions , arXiv:2504.01623v2 (2025). arxiv.org/abs/2504.01623.
  3. J. A. De Loera and B. Sturmfels, Algebraic unimodular counting , Math. Program. 96 (2003), 183–203. arXiv:math/0104286.
  4. T. Lam and A. Postnikov, Alcoved polytopes I , Discrete Comput. Geom. 38 (2007), 453–478. arXiv:math/0501246.
  5. T. Lam and A. Postnikov, Alcoved polytopes II , arXiv:1202.4015 (2012). arxiv.org/abs/1202.4015.
  6. D. Alonso-Gutiérrez, M. A. Hernández Cifre and J. Yepes Nicolás, Further inequalities for the (generalized) Wills functional , Commun. Contemp. Math. 23 (2021), 2050011. arxiv.org/abs/1912.07993.
  7. D. Iglesias, J. Yepes Nicolás and A. Zvavitch, Brunn–Minkowski type inequalities for the lattice point enumerator. Author manuscript (2019) .
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  9. J. Leake and A. H. Morales, Capacity bounds on integral flows and the Kostant partition function, arXiv:2406.07838v3 (2024). arxiv.org/abs/2406.07838.
  10. J. Tarski, The partition function for certain simple Lie algebras , Technical Report No. 7, AFOSR TN58-657 (AD162188), Department of Mathematics, University of California, Berkeley (August 1958). HathiTrust scan.

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Cite this edition

Scott Schubr. (2026). Power Concavity of Low-Dimensional Unimodular Fiber Counts (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/power-concavity-of-low-dimensional-unimodular-fiber-counts/v/2

BibTeX
@techreport{luminary_power_concavity_of_low_dimensional_unimodular_fiber_counts_v2,
  title = {Power Concavity of Low-Dimensional Unimodular Fiber Counts},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/power-concavity-of-low-dimensional-unimodular-fiber-counts/v/2}
}

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