发表机构
East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了四次Berkovich-Dhar符号变化猜想：多项式$P_n(q)^4$的系数在省略零项后符号恰好从正变负一次，并通过鞍点分析给出了转变位置的精确估计。
AI 中文摘要
设$P_n(q)=\prod_{j=1}^n(1-q^{3j-2})(1-q^{3j-1})$。我们证明，在省略零项之后，系数$[q^{3m+2}]P_n(q)^4$的符号恰好变化一次，即从正变为负。结合第二Borwein定理，这确立了Berkovich-Dhar猜想在四次情形下的符号断言。对于$n\ge 301$，该转变位于以$\alpha n^2+\beta n+\gamma+\delta/n$为中心、长度为$3000/n^2$的区间内，其中$\alpha=0.7490800947885107\ldots$，且四个常数具有显式的解析定义。证明分析了两共轭鞍点贡献之间的抵消。鞍点关系的修正控制了转变之外的系数，而高阶展开则解决了转变附近的抵消问题。在重标度变量下的一致估计覆盖了小度数范围。
英文摘要
Let $P_n(q)=\prod_{j=1}^n(1-q^{3j-2})(1-q^{3j-1})$. For each $p\in\{4,5,6,7,8\}$, we prove that the coefficients of $P_n(q)^p$ in residue class $0$ modulo $3$ are nonnegative, and that those in residue class $2$, after zero terms are omitted, change sign exactly once, from positive to negative. The residue-$0$ coefficients are in fact strictly positive for $5\le p\le8$. This proves Conjecture 2.1 of Berkovich and Dhar in full. We further determine the limiting transition constants and a four-term asymptotic expansion for the transition centres. For $n\ge301$, the signs in residue class $2$ are determined outside an interval of length $2\varepsilon_p/n^2$ centred at $α_p n^2+β_p n+γ_p+δ_p/n$, where $\varepsilon_4=1500$ and $\varepsilon_p=150$ for $5\le p\le8$, and all constants admit explicit analytic definitions. The proof combines a corrected saddle relation, higher-order expansions resolving cancellation between the two dominant saddle contributions, and a rescaled positive-integral argument that uniformly controls the small-degree range.
CommentsSubstantially expanded version. Extended from the quartic case to all powers i=4,...,8, proving Conjecture 2.1 of Berkovich and Dhar in full. 22 pages; ancillary files contain coefficient tables and transition data
