A Sharp Zero-Sector Criterion for Polynomial Prefix Sums
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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.
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Research details
- Subjects
- Mathematics · Algebra · Classical and Real Analysis
- Keywords
- Classical and Real Analysis, Algebra
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
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. The sector criterion is a universal sufficient condition. Integer lattice kernel and unitstep integral-prefix predecessors explicitly credited.
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AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work.
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The approved 6-member package contains the complete 11-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.