双旗退化与三条圆锥曲线的实切圆
Two-flag degenerations and real circles tangent to three conics
- University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
- University of Southern California（南加州大学）
- Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了定义在有理数上的强一般光滑圆锥曲线三元组，证明其有160个实切圆，反驳了至多136个的猜想，并给出了Grothendieck-Witt计数及局部符号的精确表达。
AI中文摘要：
三条一般平面圆锥曲线有$184$个复切圆，且曾有人猜想其中至多$136$个可以是实的。我们构造了一个定义在$\mathbb{Q}$上的显式强一般光滑圆锥曲线三元组，恰好有$160$个实切圆；因此相同的计数出现在一个非空欧几里得胞腔上。该构造结合了带两个旗的双线退化的四重分裂定理与精确的Sturm--Tarski、消元和区间证书。我们还证明了Grothendieck--Witt值计数为$92\mathbb{H}$，并在固定定向约定下，用曲率差、剩余交除子和接触法线表达其实局部符号。精确算术代码和证书数据存档于随附的存储库中。
英文摘要：
Three general plane conics admit $184$ complex tangent circles, and it had been conjectured that at most $136$ of them could be real. We construct an explicit strongly general triple of smooth conics over $\mathbb{Q}$ with exactly $160$ real tangent circles; the same count therefore occurs on a nonempty Euclidean chamber. The construction combines a fourfold splitting theorem for two flagged double-line degenerations with exact Sturm--Tarski, elimination, and interval certificates. We also show that the Grothendieck--Witt-valued count is $92\mathbb{H}$ and express its real local signs, up to a fixed orientation convention, in terms of curvature differences, residual intersection divisors, and contact normals. The exact-arithmetic code and certificate data are archived in the accompanying repository.
