# Sum-Product Phenomena for Planar Hypercomplex Numbers

## TL;DR

This paper investigates the sum-product problem in planar hypercomplex numbers, specifically dual and double numbers, revealing their unique combinatorial behaviors and extending incidence geometry techniques.

## Contribution

It introduces new sum-product bounds for hypercomplex numbers and develops analogs of the Szemeredi-Trotter theorem for these systems.

## Key findings

- Sum-product bounds depend on specific parameters of hypercomplex systems.
- Dual numbers exhibit a range where sum and product growth are balanced.
- Extended incidence bounds differ from classical real and complex cases.

## Abstract

We study the sum-product problem for the planar hypercomplex numbers: the dual numbers and double numbers. These number systems are similar to the complex numbers, but it turns out that they have a very different combinatorial behavior. We identify parameters that control the behavior of these problems, and derive sum-product bounds that depend on these parameters. For the dual numbers we expose a range where the minimum value of $\max\{|A+A|,|AA|\}$ is neither close to $|A|$ nor to $|A|^2$. To obtain our main sum-product bound, we extend Elekes' sum-product technique that relies on point-line incidences. Our extension is significantly more involved than the original proof, and in some sense runs the original technique a few times in a bootstrapping manner. We also study point-line incidences in the dual plane and in the double plane, developing analogs of the Szemeredi-Trotter theorem. As in the case of the sum-product problem, it turns out that the dual and double variants behave differently than the complex and real ones.