# On Rado conditions for nonlinear Diophantine equations

## TL;DR

This paper extends Rado's conditions to nonlinear Diophantine equations, proposing necessary conditions for partition regularity and verifying them for specific quadratic equations, aiming to generalize Rado's theorem.

## Contribution

It introduces necessary conditions inspired by Rado's theorem for nonlinear equations and verifies their sufficiency in certain quadratic cases, advancing the understanding of partition regularity.

## Key findings

- Necessary conditions for nonlinear Diophantine equations' partition regularity.
- Verification of the conjecture for specific quadratic equations with certain parameter constraints.
- New results on partition regularity of polynomial configurations in integers.

## Abstract

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado's characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado's theorem. We verify that such a conjecture hold for the equations $x^{2}-xy+ax+by+cz=0$ and $x^{2}-y^{2}+ax+by+cz=0$ for $a,b,c\in \mathbb{Z}$ such that $abc=0$ or $% a+b+c=0$. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in $\mathbb{Z}$ such as $\left\{ x,x+y,xy+x+y\right\} $, building on the recent result on the partition regularity of $\left\{ x,x+y,xy\right\} $.