Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited

Abstract

Let F ∈ C[x, y, z] be a constant-degree polynomial, and let A, B, C ⊂ C be finite sets of size n. We show that F vanishes on at most O(n 11/6 ) points of the Cartesian product A × B × C, unless F has a special group-related form. This improves a theorem of Elekes and Szabó [4], and generalizes a result of Raz, Sharir, and Solymosi [13]. The same statement holds over R, and a similar statement holds when A, B, C have different sizes (with a more involved bound replacing O(n 11/6 )).This result provides a unified tool for improving bounds in various Erdős-type problems in combinatorial geometry, and we discuss several applications of this kind.

The Elekes–Szabó Theorem in four dimensions

“…Elekes and Rónyai [3] and Elekes and Szabó [4] pioneered the study of algebraic structures behind problems from combinatorial geometry. The main result of [4] was quantitatively improved by the authors in [15] to the following statement; we state it here in a somewhat rough form, and refer to [15] for a full and precise statement. Given an irreducible polynomial F ∈ C[x, y, z] and finite sets A, B, C ⊂ C of size n, we have the bound (writing Z(F ) for the zero set of F )…”

Constructions for the Elekes–Szabó and Elekes–Rónyai problems

“…Elekes and Szabó [6] showed that if the polynomial is not degenerate in this sense, then the bound (1) can be improved to n 2−η for some η > 0. A quantitative improvement to η = 1/6 was obtained by Raz, Sharir and de Zeeuw [9], leading to the following statement. 6,9]).…”

“…A quantitative improvement to η = 1/6 was obtained by Raz, Sharir and de Zeeuw [9], leading to the following statement. 6,9]). Let F ∈ R[x, y, z] be a polynomial of degree d. If F is not degenerate, then for any A, B, C ⊂ R of size n we have…”

Expanding Polynomials on Sets With Few Products

“…Theorem generalizes many problems from discrete geometry and additive combinatorics, and so has many applications (for example, see or ). While we will not aim to give a complete background, it is important to also mention that the analogous problem has been considered over different fields instead of , where many interesting results are also available.…”

“…Let d be a positive integer, let A, B ⊂ R be finite sets, and let f ∈ R[x, y] be of degree d. Then, unless f = h(g 1 (x) + g 2 (y)) or f = h(g 1 (x) · g 2 (y)) for some h, g 1 , g 2 ∈ R[x], we have f (A, B) = min |A| 2/3 |B| 2/3 , |A| 2 , |B| 2 . Theorem 1.1 generalizes many problems from discrete geometry and additive combinatorics, and so has many applications (for example, see [10] or [11]). While we will not aim to give a complete background, it is important to also mention that the analogous problem has been considered over different fields instead of R, where many interesting results are also available.…”