平面点集笛卡尔积上多项式的消元检验：共线性检验及相关问题
Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related Problems
摘要：
我们提出了次二次算法，在代数决策树模型的计算，用于检测是否存在一个三元组的点，属于三个各自的集合$A$，$B$，和$C$的点在平面上，满足一定的多项式方程或两个方程。最著名的例子是测试共线三重点的存在$A\times B\times C$，一个经典的3SUM困难的问题，迄今为止，任何试图获得一个次二次解决方案，无论是在（均匀）真正的RAM模型，或在代数决策树模型。虽然我们仍然无法解决这个问题，在充分的一般性，在次二次时间，我们得到这样一个解决方案，在代数决策树模型，仅使用大致$O(n^{28/15})$常数次数多项式符号测试，为特殊情况下，其中两个集位于两个各自的一维曲线和第三个是任意放置在平面上。我们的技术是相当普遍的，并适用于许多其他问题，我们寻求一个三元组，满足一个单一的多项式方程，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~ $C$都被假设为一维的。作为我们技术的第二个应用，我们在平面上又有三个$n$ -点集$A$、$B$和$C$，我们想确定是否存在同时满足两个独立实多项式方程的三元组$(a,b,c) \in A\times B\times C$。例如，这是当测试复平面中的共线性时的设置，此时集合$A$、$B$、$C$中的每一个都位于某个常次数代数曲线上。我们表明，这种问题可以解决大致$O(n^{24/13})$常数度多项式符号测试。
英文摘要：
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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