二元形式的Julia零点与双曲零点的平衡律

中文摘要 AI 辅助

对于无实根的实二元形式，Julia零点与双曲零点是上半平面中两个SL(2,R)等变点。我们证明二者具有密切相关的平衡特征，其径向权重分别由到根的距离的双曲正切与双曲正弦给出。该共同框架为两个零点映射的重合性提供了几何判据：对于二元四次式，二者始终一致；对于二元六次式，当三个上半平面根构成等边双曲三角形，或与其中一个根共线且该根为另两个根的双曲中点时，二者完全一致；我们还得到了共线二元八次式的对应结果。两个平衡律进一步揭示了远根影响的显著差异：局限于紧集的严格多数根可将Julia零点保持在紧集内，且1/2的阈值是最优的；相反，逃逸的少数根可迫使双曲零点以线性尺度逃逸。我们还阐明了双曲零点显式公式的计算优势。

英文摘要

For a real binary form with no real roots, Julia's zero and the hyperbolic zero are two SL(2,R)-equivariant points in the upper half-plane. We show that they admit closely related equilibrium characterizations, with radial weights given respectively by the hyperbolic tangent and hyperbolic sine of the distances to the roots. This common framework gives geometric criteria for coincidence of the two zero maps. They always agree for binary quartics; for binary sextics they agree exactly when the three upper-half-plane roots form an equilateral hyperbolic triangle, or are collinear with one root the hyperbolic midpoint of the other two. We also obtain a corresponding result for collinear binary octics. The two equilibrium laws further reveal a sharp difference in the influence of distant roots. A strict majority of roots confined to a compact set keeps Julia's zero in a compact set, and the threshold one-half is optimal. In contrast, an escaping minority can force the hyperbolic zero to escape at linear scale. We also illustrate the computational advantages of the explicit formula for the hyperbolic zero.

发表机构

• American University（美国大学）

机构由 AI 辅助整理，请以论文原文为准。