We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets $A$, $B$, and $C$ of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in $A\times B\times C$, a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses only roughly $O(n^{28/15})$ constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to many other problems where we seek a triple that satisfies a single polynomial equation, e.g., determining whether $A\times B\times C$ contains a triple spanning a unit-area triangle. This result extends recent work by Barba \etal~(2017) and by Chan (2018), where all three sets $A$,~$B$, and~$C$ are assumed to be one-dimensional. As a second application of our technique, we again have three $n$-point sets $A$, $B$, and $C$ in the plane, and we want to determine whether there exists a triple $(a,b,c) \in A\times B\times C$ that simultaneously satisfies two independent real polynomial equations. For example, this is the setup when testing for collinearity in the complex plane, when each of the sets $A$, $B$, $C$ lies on some constant-degree algebraic curve. We show that problems of this kind can be solved with roughly $O(n^{24/13})$ constant-degree polynomial sign tests.

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测试多项式在平面点集的笛卡尔积上消失：共线性测试和相关问题

我们提出了在代数决策树计算模型中的次二次算法，用于检测是否存在属于平面中三个各自的点集合 $A$、$B$ 和 $C$ 的三元组点，满足某个多项式方程或两个方程。此类问题最著名的例子是测试 $A\times B\times C$ 中是否存在共线三元组点，这是一个经典的 3SUM 难题，迄今为止，任何试图获得次二次解的尝试都失败了，无论是在（统一的）真实 RAM 模型中，或在代数决策树模型中。虽然我们仍然无法解决这个问题，但总的来说，在次二次时间，我们在代数决策树模型中获得了这样一个解决方案，它仅使用大约 $O(n^{28/15})$ 常数-多项式符号测试，对于特殊情况，其中两个集合位于两个各自的一维曲线上，而第三个集合任意放置在平面中。我们的技术相当通用，适用于许多其他问题，其中我们寻求满足单个多项式方程的三元组，例如，确定 $A\times B\times C$ 是否包含跨越单位面积三角形的三元组。这一结果扩展了 Barba \etal~(2017) 和 Chan (2018) 最近的工作，其中所有三个集合 $A$、~$B$ 和~$C$ 都被假定为一维的。作为我们技术的第二个应用，我们再次在平面上有三个 $n$-点集 $A$、$B$ 和 $C$，我们想确定是否存在三元组 $(a,b, c) 同时满足两个独立的实数多项式方程的\in A\times B\times C$。例如，这是在复平面中测试共线性时的设置，当每个集合 $A$、$B$、$C$ 位于某个常次代数曲线上时。我们表明，这类问题可以通过大约 $O(n^{24/13})$ 常数次多项式符号测试来解决。