  The Casas-Alvero conjecture (2001) predicts that the $n^{th}$ "arithmetic
Casas-Alvero scheme" $X_n\rightarrow \operatorname{Spec}(\mathbb{Z})$
(constructed by von Bothmer-Labs-Schicho-de Woestline) does not have any
$\mathbb{K}$-rational points for any characteristic zero field $\mathbb{K}$ and
$n\geq 2$. In this paper, we show that for any field $\mathbb{K}$ (of any
characteristic), $X_n$ has at most finitely many $\mathbb{K}$-rational points
for all $n\geq 2$. We reformulate the conjecture as a complete intersection
problem, which enables us to interpret the finiteness of rational points on
$X_n$, as an almost complete intersection phenomenon. Furthermore, we prove
several rigidity results towards the conjecture.
