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A Sharp Zero-Sector Criterion for Polynomial Prefix Sums

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Classical and Real Analysis
  • Algebra
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

Let h be a real polynomial with h(0)>0 whose zeros lie in the closed sector of half-angle pi/4 around the negative real axis. For every nonnegative real step and geometric bias at least one, its geometrically weighted integer prefix sums admit a finite concave logarithmic extension to the nonnegative quadrant. This implies weighted logarithmic Jensen inequalities for arbitrary probability laws with finite first moments and integral mean prefix length. The universal sector angle and bias range are sharp, including integral-height counterexamples. Joint log-concavity of a two-variable truncated-square lattice sum follows from complete geometry coverage and exact rational certificates; classical Prekopa marginalization removes the degree bound. Applications cover split projective-line bundles over projective space in every dimension. The extension agrees with integer-prefix rows rather than asserting ordinary polynomial continuation in the prefix variable.

Keywords Classical and Real Analysis · Algebra

Scope of the work

Let h be a real polynomial with h(0)>0 whose zeros lie in the closed sector of half-angle pi/4 around the negative real axis. For every nonnegative real step and geometric bias at least one, its geometrically weighted integer prefix sums admit a finite concave logarithmic extension to the nonnegative quadrant. This implies weighted logarithmic Jensen inequalities for arbitrary probability laws with finite first moments and integral mean prefix length. The universal sector angle and bias range are sharp, including integral-height counterexamples. Joint log-concavity of a two-variable truncated-square lattice sum follows from complete geometry coverage and exact rational certificates; classical Prekopa marginalization removes the degree bound. Applications cover split projective-line bundles over projective space in every dimension. The extension agrees with integer-prefix rows rather than asserting ordinary polynomial continuation in the prefix variable. Universal sufficient sector is not individual necessity; no arbitrary SLC input, entire-function, arbitrary toric/unimodular, ordinary polynomial-continuation or individual zero-sector necessity assertion. Integer lattice kernel and unitstep integral-prefix predecessors explicitly credited.

Reproduction notes

The approved 6-member package contains the complete 12-page paper, editable LaTeX sources and dependencies, reader/build instructions, scholarly references and a public checksum manifest, with necessary exact verification programs and certificates. Follow the included README and retain the package together. Exact finite certificates support the stated analytic argument; they do not certify external human peer review.

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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.

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Cite this edition

Scott Schubr. (2026). A Sharp Zero-Sector Criterion for Polynomial Prefix Sums (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/a-sharp-zero-sector-criterion-for-polynomial-prefix-sums/v/1

BibTeX
@techreport{luminary_a_sharp_zero_sector_criterion_for_polynomial_prefix_sums_v1,
  title = {A Sharp Zero-Sector Criterion for Polynomial Prefix Sums},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/a-sharp-zero-sector-criterion-for-polynomial-prefix-sums/v/1}
}

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