Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related Problems

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.