It is well known that a polynomial $f \in K[x_1, \ldots, x_n]$ is translation invariant iff $Df = 0$ where $$D = \displaystyle\frac{\partial}{\partial x_1} + \ldots + \frac{\partial}{\partial x_n}$$ I would need a reference for this fact. Skimming through Dolgachev, I couldn't find it.
$\textbf{EDIT}$: The ring $R = K[x_1, \ldots, x_n]$ comes naturally equipped with a $\mathbb{G}_a$ action given by $$t.x = x + t(1, \ldots, 1)$$ A translation invariant is then an element of $R^{\mathbb{G}_a}$.
What you've stated is false, e.g. consider the polynomial $p(x_1,x_2):=x_1-x_2$. $p$ is not translation invariant, however $Dp(x_1,x_2)=1-1 = 0$.
Presumably you want to say $p$ is translation invariant iff $\frac{\partial}{\partial x_i}p\equiv 0$ for all $i$? In which case, I suggest you write down the definitions clearly and go from there! Translation invariance means $$p(x_1+h_1,\dots, x_n+h_n)=p(x_1,\dots,x_n)$$ for all $\mathbf{x, h}$ and $$\partial_{x_i}p(x)=\lim_{\delta\to 0}\frac{p(x_1,\dots,x_i+\delta,\dots,x_n)-p(x_1,\dots,x_n)}{\delta}.$$