Duke Mathematical Journal
- Duke Math. J.
- Volume 165, Number 18 (2016), 3517-3566.
Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited
Let be a constant-degree polynomial, and let be finite sets of size . We show that vanishes on at most points of the Cartesian product , unless has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over , and a similar statement holds when have different sizes (with a more involved bound replacing ). 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.
Duke Math. J., Volume 165, Number 18 (2016), 3517-3566.
Received: 28 April 2015
Revised: 18 February 2016
First available in Project Euclid: 7 September 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Raz, Orit E.; Sharir, Micha; De Zeeuw, Frank. Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited. Duke Math. J. 165 (2016), no. 18, 3517--3566. doi:10.1215/00127094-3674103. https://projecteuclid.org/euclid.dmj/1473275860