The Elekes–Szabó Theorem in four dimensions

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 ( n 8/3 ) 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, with a suitably modified bound on the number of zeros. 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.