一般六次方程解集的几何结构
The geometry of solutions to the general sextic
AI总结：
本文研究一般六次方程根的三个最简公式的几何，证明其标准形式无同时一致化，揭示六次方程特有的定性复杂性，并给出基于Hilbert模尖点形式的根公式。
AI中文摘要：
我们研究了一般六次多项式根的三个最简已知公式的几何结构。与Green关于五次方程的工作形成对比，我们证明了与这些解相关的标准形式不存在“同时一致化”。这表明六次方程具有五次方程所不具备的内在定性复杂性。此外，我们给出了用Hilbert模尖点形式及相应Hilbert模曲面的一致化微分方程来表示一般六次方程根的公式。
英文摘要：
We study the geometry of the three simplest known formulas for roots of a general sextic polynomial. In contrast with work of Green on the quintic, we show that no "simultaneous uniformization" of the normal forms associated to the solutions exists. This demonstrates an intrinsic qualitative complexity of sextic equations that is not present in quintic equations. Additionally, we give a formula for the roots of a general sextic in terms of Hilbert modular cusp forms and the uniformizing differential equation of the associated Hilbert modular surface.
