# Constructions for the Elekes-Szab\'o and Elekes-R\'onyai problems

## TL;DR

This paper constructs specific polynomials and point sets to establish new lower bounds for the Elekes-Szabó and Elekes-Rónyai problems, advancing understanding of their combinatorial complexity.

## Contribution

It provides the first non-degenerate polynomial construction with large intersection with a grid, improving lower bounds for these problems.

## Key findings

- Constructed a non-degenerate polynomial with intersection size
n^{3/2}
- Developed a polynomial f not of additive or multiplicative form with many points where f takes limited values
- Established new lower bounds for the Elekes-Szabf3 and Elekes-Rf3nyai problems

## Abstract

We give a construction of a non-degenerate polynomial $F\in \mathbb R[x,y,z]$ and a set $A$ of cardinality $n$ such that $\left|Z(F)\cap (A \times A \times A) \right| \gg n^{\frac{3}{2}}$, thus providing a new lower bound construction for the Elekes--Szab\'o problem. We also give a related construction for the Elekes--R\'onyai problem restricted to a subgraph. This consists of a polynomial $f\in \mathbb R[x,y]$ that is not additive or multiplicative, a set $A$ of size $n$, and a subset $P\subset A\times A$ of size $|P|\gg n^{3/2}$ on which $f$ takes only $n$ distinct values.