Constructions for the Elekes-Szabó and Elekes-Rónyai problems

2018/12/03 by Makhul, Mehdi, Roche-Newton, Oliver, Warren, Audie +1
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.00654

Abstract

We give a construction of a non-degenerate polynomial F∈ \mathbb R[x,y,z] and a set A of cardinality n such that |Z(F)∩ (A × A × A) | ≫ n^(3)/(2), thus providing a new lower bound construction for the Elekes--Szabó problem. We also give a related construction for the Elekes--Rónyai problem restricted to a subgraph. This consists of a polynomial f∈ \mathbb R[x,y] that is not additive or multiplicative, a set A of size n, and a subset P⊂ A× A of size |P|≫ n^3/2 on which f takes only n distinct values.

Related

• The Elekes-Szabó Problem and the Uniformity Conjecture
• A symmetric multivariate Elekes-Rónyai theorem
• Improved bound for the k-variate Elekes--Rónyai theorem
• A survey of Elekes-Rónyai-type problems
• An Elekes-Rónyai theorem for sets with few products