Log-concave polynomial prefix sums: algebraic certificates and a sharp zero sector

Abstract

We give algebraic certificates for joint logarithmic concavity of polynomial prefix sums H_k(y)=∑(i=0,…,k) ρⁱ h(y+ai), for every real step a ≥ 0 and bias ρ ≥ 1. It suffices that the factorial coefficients of h are nonnegative and log-concave with interval support, that h(0)>0, and that h(U)h(V) has a nonnegative expansion in UV and U+V. The resulting class is closed under products, positive affine changes of variable and nonzero derivatives. It contains all real polynomials with zeros in the closed sector |arg(-z)| ≤ π/4 and strictly more, including an explicit cubic outside that sector. The logarithmic extension agrees exactly at integer prefix lengths and gives Jensen inequalities for arbitrary finite-first-moment laws with integral mean prefix length. The sector angle and universal bias range are sharp. The analytic ingredient is a real two-variable truncated-square kernel, proved by complete chamber formulas and exact rational polynomial certificates. Classical Prékopa marginalization gives its sharp real smoothing threshold. An application covers split projective-line bundles over projective space in every dimension.