Research Article · Preprint Mathematics
A sharp zero-sector criterion for log-concave polynomial prefix sums
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis · Algebra
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
Let h be a real polynomial with h(0)>0 whose zeros lie in the closed sector |arg(-z)|≤π/4. For every real step a≥0 and geometric bias ρ≥1, we prove that H_k(y)=∑_{i=0}^kρ^i h(y+ai) admits a finite concave logarithmic extension from k∈ℕ, y≥0 to the nonnegative quadrant. This implies weighted logarithmic Jensen inequalities for arbitrary probability laws with finite first moments and integral mean prefix length. The sector angle and the universal range ρ≥1 are sharp, including counterexamples with integral heights. The main analytic ingredient is joint log-concavity of a two-variable truncated-square lattice sum. Four complete geometry cases and exact rational polynomial certificates establish that ingredient; classical Prékopa marginalization then removes the degree bound and yields a sharp real smoothing threshold. The resulting application covers split projective-line bundles over projective space in every dimension. The conclusion concerns a concave extension of integer-prefix rows, rather than an ordinary polynomial continuation in the prefix variable.
Keywords Classical and Real Analysis · Algebra
Scope of the work
A sharp universal zero-sector criterion giving exact concave logarithmic extensions of polynomial prefix rows and integral-mean Jensen inequalities, with a sharp real smoothing range.
Reproduction notes
12-page manuscript and 6 package members. Compile main.tex with compatible LaTeX or Tectonic. Run python -B verify_prefix.py and python -O -B verify_prefix.py using the Python 3 standard library. Both reconstruct the complete chamber, determinant, moment, sector and sharpness identities and print identical JSON without external inputs.
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Author disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). A sharp zero-sector criterion for log-concave polynomial prefix sums (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/a-sharp-zero-sector-criterion-for-polynomial-prefix-sums/v/2
BibTeX
@techreport{luminary_a_sharp_zero_sector_criterion_for_polynomial_prefix_sums_v2,
title = {A sharp zero-sector criterion for log-concave polynomial prefix sums},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/a-sharp-zero-sector-criterion-for-polynomial-prefix-sums/v/2}
}Loading discussion…