A proof of the quartic Berkovich-Dhar sign-change conjecture

Source abstract

Let $P_n(q)=\prod_{j=1}^n(1-q^{3j-2})(1-q^{3j-1})$. We prove that, after zero terms are omitted, the coefficients $[q^{3m+2}]P_n(q)^4$ change sign exactly once, from positive to negative. Together with the Second Borwein Theorem, this establishes the sign assertions in the quartic case of the Berkovich-Dhar conjecture. For $n\ge 301$, the transition lies in an interval of length $3000/n^2$ centred at $αn^2+βn+γ+δ/n$, where $α=0.7490800947885107\ldots$ and the four constants have explicit analytic definitions. The proof analyses the cancellation between two conjugate saddle contributions. A correction to the saddle relation controls the coefficients away from the transition, while a higher-order expansion resolves the cancellation near it. Uniform estimates in a rescaled variable cover the small-degree range.