AI 中文总结
该文针对Reznick提出的平面集五点子集确定圆锥曲线的问题，证明p>2时幂曲线与近顶点的拉梅卵形线五点子集可确定非退化双曲线，修正了p=2.001时拉梅卵形线仅产生椭圆的猜想。
AI 中文摘要
Croft--Falconer--Guy的问题A33记录了Reznick关于无限平面集的问题：每个五点子集都能确定椭圆或双曲线，并提出|x|^2.001 + |y|^2.001 = 1可能仅产生椭圆。本文给出两个结果：第一，若p>2且a>0，幂曲线u=ay^p（y>0）上任意五个不同点确定非退化双曲线，因此有界子弧为Reznick问题的双曲线侧提供了非圆锥的可求长实例；第二，每个固定的单侧五点轮廓向拉梅卵形线|x|^p + |y|^p = 1的轴向顶点收缩时，最终确定非退化双曲线。由此，p=2.001时的拉梅卵形线存在确定非退化双曲线的五点子集，并非仅产生椭圆，但跨越同一顶点的对称轮廓确定椭圆。本文还将这些结果与等仿射曲率提供的经典密切圆锥准则区分开。
英文摘要
Problem A33 of Croft--Falconer--Guy records Reznick's question about infinite plane sets for which every five-point subset determines an ellipse or, respectively, a hyperbola, and suggests that $|x|^{2.001}+|y|^{2.001}=1$ might yield only ellipses. We give two results. First, if $p>2$ and $a>0$, every five distinct points of the power curve $u=ay^p$, $y>0$, determine a nondegenerate hyperbola; bounded subarcs therefore give nonconic rectifiable examples for the hyperbolic side of Reznick's question. Second, every fixed one-sided five-point profile, contracted toward an axial vertex of the Lamé oval $|x|^p+|y|^p=1$, eventually determines a nondegenerate hyperbola. Consequently, the suggested Lamé oval with $p=2.001$ has five-point subsets that determine nondegenerate hyperbolas, so it does not yield only ellipses. Symmetric profiles straddling the same vertex, however, determine ellipses. We also separate these results from the classical osculating-conic criterion supplied by equi-affine curvature.
