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, 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.
Bibliographical noteFunding Information:
Work on this paper by Orit E. Raz and Micha Sharir was supported by Grant 892/13 from the Israel Science Foundation, and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by Orit E. Raz was also supported by a Shulamit Aloni Fellowship from the Israeli Ministry of Science. Work by Micha Sharir was also supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work on this paper by Frank de Zeeuw was partially supported by Swiss National Science Foundation Grants 200020-165977 and 200021-162884. Received November 1, 2016 and in revised form August 3, 2017
© 2018, Hebrew University of Jerusalem.