On Rado conditions for nonlinear Diophantine equations

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\} $.