A sharp zero-sector criterion for log-concave polynomial prefix sums

Scott Schubr

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 extension agrees exactly with the logarithm on every integer-prefix row. The construction treats real step sizes directly and preserves agreement with the original discrete prefix sums.

A sharp zero-sector criterion for log-concave polynomial prefix sums

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