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  one-dimensional curves and the third is placed arbitrarily in the plane. \etal 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  real polynomial equations. For example, this is the setup when testing for  the existence of pairs of similar triangles spanned by the input points, in various contexts discussed later in the paper . We show that problems of this kind can be solved with roughly O(n^{24/13}) constant-degree polynomial sign tests . This is a considerable improvement, given the 3SUM-hardness of the triangle similarity problem that we discuss and the lack of any subquadratic algorithm for it (or for any similar problem) in the uniform model .
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 .
A
two respective
A
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
A
~(2017) and by Chan (2018), where all three sets A,~B, and~C are assumed to be one-dimensional.
A
independent
A
collinearity in the complex plane, when each of
R
existence of pairs of similar triangles spanned by the input points, in various contexts discussed later in the paper
sets A, B, C lies on some constant-degree algebraic curve
D
. This is a considerable improvement, given the 3SUM-hardness of the triangle similarity problem that we discuss and the lack of any subquadratic algorithm for it (or for any similar problem) in the uniform model