The Elekes–Szabó Theorem in four dimensions

Orit E. Raz, Micha Sharir, Frank de Zeeuw*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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 languageAmerican English
Pages (from-to)663-690
Number of pages28
JournalIsrael Journal of Mathematics
Issue number2
StatePublished - 1 Aug 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2018, Hebrew University of Jerusalem.


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