The Elekes–Szabó Theorem in four dimensions

Orit E. Raz, Micha Sharir, Frank de Zeeuw

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)663-690
Number of pages28
JournalIsrael Journal of Mathematics
Volume227
Issue number2
DOIs
StatePublished - 1 Aug 2018

Fingerprint

Dive into the research topics of 'The Elekes–Szabó Theorem in four dimensions'. Together they form a unique fingerprint.

Cite this