Locating Quintic Obstructions in the Riemann Theta Product: A certified transition and minimum-degree theorem

Overview

Abstract

We give a reproducible method for locating and certifying quintic obstructions for the normalized fixed-center Riemann theta product, and prove a unique transition in its odd quintic moment block on [3.64, 3.66]. The transition parameter lies strictly between 3.64888758188562 and 3.64888758188563. Within this interval, below the transition, degree five is the minimum degree of a polynomial whose square has negative signed-moment value; at the transition the degree-five form is positive semidefinite with a one-dimensional nullspace; above it the form is positive definite. The proof combines a Schur-complement reduction with uniform directed interval bounds, including the complete infinite theta tails. An exact rational arithmetic replay verifies the bounds on the entire interval. An explicit integer-coefficient quintic illustrates why testing one fixed polynomial does not locate the transition of the full quintic block.

Keywords Riemann theta density · moment matrices · positive definiteness · quintic obstruction · computer-assisted proof