发表机构
School of Mathematical Sciences, Guangxi Minzu University（广西民族大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文构造了一个实有理系数的显式首一多项式，其七个单零点在单位圆盘内，并用认证分隔线证明连接零点的路径长度大于2+10^-8，提供了数值验证的反例。
AI 中文摘要
在先前报道的ani的七次反例的背景下，我们提出了一个具有实有理系数的显式首一多项式，并通过认证分隔线给出了直接证明。该多项式在开单位圆盘内有七个单零点。在其闭单位下水平集中连接不同零点的每条连续路径，其像的一维豪斯多夫测度大于$2+10^{-8}$。五条分段线性图排除了二十一零点对中的二十对；两条短的禁止线段迫使剩余一对进行定量绕行。精确的有理伯恩斯坦证书在四十四个终端区间及其无界尾部上验证了图不等式。该证明不需要临界点计数或解析逆分支。我们单独记录了将路径长度障碍转移到豪斯多夫测度和闭下水平集的一般连续统与膨胀论证。这些转移并非作为新原理提出。重点在于显式的实系数构造、其分隔证书及其验证的数值余量，而非对长度二问题的首个否定答案或对每个更大普适界限的反驳。
英文摘要
In the setting of the previously reported degree-seven counterexample of ani, we present an explicit monic polynomial with real rational coefficients and a direct proof by certified separators. The polynomial has seven simple zeros in the open unit disk. Every continuous path joining distinct zeros in its closed unit sublevel set has image of one-dimensional Hausdorff measure greater than $2+10^{-8}$. Five piecewise-linear graphs exclude twenty of the twenty-one zero pairs; two short forbidden segments force a quantitative detour for the remaining pair. Exact rational Bernstein certificates verify the graph inequalities on forty-four terminal intervals and their unbounded tails. The proof does not require critical-point counting or analytic inverse branches. We separately record the general continuum and dilation arguments that transfer path-length obstructions to Hausdorff measure and to closed sublevel sets. Those transfers are not asserted as new principles. The focus is the explicit real-coefficient construction, its separation certificates, and its verified numerical margin, rather than a first negative answer to the length-two question or a disproof of every larger universal bound.
