Subquadratic Algorithms for Algebraic 3SUM

The 3SUM problem asks if an input n -set of real numbers contains a triple whose sum is zero. We qualify such a triple of degenerate because the probability of finding one in a random input is zero. We consider the 3POL problem, an algebraic generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz et al. gave an O ( n 11 / 6 ) upper bound on the number of degenerate triples for the 3POL problem. We give algorithms for the corresponding problem of counting them. Grønlund and Pettie designed subquadratic algorithms for 3SUM. We prove that 3POL admits bounded-degree algebraic decision trees of depth O ( n 12 / 7 + ε ) , and we prove that 3POL can be solved in O ( n 2 ( log log n ) 3 / 2 / ( log n ) 1 / 2 ) time in the real-RAM model, generalizing their results. Finally, we shed light on the General Position Testing (GPT) problem: “Given n points in the plane, do three of them lie on a line?”, a key problem in computational geometry: we show how to solve GPT in subquadratic time when the input points lie on a small number of constant-degree polynomial curves. Many other geometric degeneracy testing problems reduce to 3POL.