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 holds 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 $\left\{x,x+y,xy\right\}$.

在Di Nasso和Luperi Baglini的先前工作的基础上，我们提供了Diophantine方程成为分区正规的一般必要条件。这些条件是受Rado对分区正则线性齐次方程的刻画所启发。我们推测，这些条件也足以满足分配规则性，至少对于其单变量多项式为线性的方程式而言已经足够。这将提供拉多定理的自然概括。

我们验证了这样的猜想对等式成立 $X^2-Xÿ+\mathrm{一种}X+bÿ+Cž=0$ 和 $X^2-{ÿ}^2+\mathrm{一种}X+bÿ+Cž=0$ 对于 $\mathrm{一种}，b，C\in \mathbb{ž}$ 这样 $\mathrm{一种}bC=0$ 要么 $\mathrm{一种}+b+C=0$。为了处理这些方程，我们建立了关于多项式配置的分区正则性的新结果。$\mathbb{ž}$ 如 $\left\{X，X+ÿ，Xÿ+X+ÿ\right\}$，基于最近的分区规则性结果 $\left\{X，X+ÿ，Xÿ\right\}$。