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Power Concavity of Low-Dimensional Unimodular Fiber Counts

Scott Schubr

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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. 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.

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Edition files

Available editions

  1. Edition 5 · 2026-11-02 · public
  2. Edition 4 · 2026-10-27 · public
  3. Edition 3 · 2026-10-24 · public
  4. Edition 2 · 2026-10-24 · public
  5. Edition 1 · 2026-10-22 · public

Research details

Subjects
Mathematics · Combinatorics and Graph Theory
Keywords
unimodular fibers, integer flows, power concavity, lattice-point counts, Kostant partition function, Brunn–Minkowski inequality
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Weighted Jensen inequalities apply under the stated bounded unimodular-fiber and central-dimension hypotheses. The continuous type-A₃ exponent 4/13 is optimal, and bounds are provided for the lattice-only exponent.

Disclosures

AI-assisted tools were used in developing and internally checking the arguments and preparing the manuscript. Classical convex geometry and chamber enumeration are explicitly credited.

Reproduction

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.

References and related resources

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