2016/07/13 by Raz, Orit E., Sharir, Micha, de Zeeuw, Frank · 1 citation
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.1607.03600
Let F∈ℂ[x,y,s,t] be an irreducible constant-degree polynomial, and let A,B,C,D⊂ℂ be finite sets of size n. We show that F vanishes on at most O(n8/3) points of the Cartesian product A× B× C× D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes. This is a four-dimensional extension of our recent improved analysis of the original Elekes-Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.