  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\} $.
