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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 subfield
Combinatorics and Graph Theory
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · 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. 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

The corrected square-root and count-dependent power inequalities assume bounded nonempty unimodular fibers with central affine dimension at most two and admissible finite weighted integral demands. The leading asymptotic deficit is sharp; exact optimal powers at finite thresholds remain undetermined. The 4/13 optimality result applies to the specified classical continuous type-A₃ extension, not to the best exponent restricted to integer demands. The unrestricted higher-dimensional Jensen question remains open.

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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).
  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).
  3. J. A. De Loera and B. Sturmfels, Algebraic unimodular counting, Mathematical Programming 96 (2003), 183–203.
  4. T. Lam and A. Postnikov, Alcoved polytopes I, Discrete & Computational Geometry 38 (2007), 453–478.
  5. T. Lam and A. Postnikov, Alcoved polytopes II, arXiv:1202.4015 (2012).
  6. D. Alonso-Gutiérrez, M. A. Hernández Cifre and J. Yepes Nicolás, Further inequalities for the (generalized) Wills functional, Communications in Contemporary Mathematics 23 (2021), 2050011.
  7. D. Iglesias, J. Yepes Nicolás and A. Zvavitch, Brunn–Minkowski type inequalities for the lattice point enumerator, author manuscript (2019).
  8. A. Barvinok, Brunn–Minkowski inequalities for contingency tables and integer flows, Advances in Mathematics 211 (2007), 105–122.
  9. J. Leake and A. H. Morales, Capacity bounds on integral flows and the Kostant partition function, arXiv:2406.07838v3 (2024).
  10. J. Tarski, The partition function for certain simple Lie algebras, Technical Report No. 7, AFOSR TN58-657 (AD162188), University of California, Berkeley (August 1958).

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Power Concavity of Low-Dimensional Unimodular Fiber Counts (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/power-concavity-of-low-dimensional-unimodular-fiber-counts/v/1

BibTeX
@techreport{luminary_power_concavity_of_low_dimensional_unimodular_fiber_counts_v1,
  title = {Power Concavity of Low-Dimensional Unimodular Fiber Counts},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/power-concavity-of-low-dimensional-unimodular-fiber-counts/v/1}
}

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