First, some teasers; then, at the end of 2), the question.
The above statement holds for polynomials which are normalised (the leading coefficient is one, or a unit) and defined in rings $R[X]$ where $R$ a commutative ring.
1) If your polynomial is not normalised then there can be an infinity of solutions, of course. Take, for instance, $P_\phi=\phi X$ over the ring $R=\mathbb F_2^E$ of functions on some infinite set $E$ into the ring $\mathbb F_2 = \mathbb Z/2\mathbb Z$ and let $\phi \in R$ be such that $\phi^{-1}(\{1\})$ is finite. Note however, that even for those $\phi$ for which $P_\phi$ has only finitely many solutions, it can be more than $1 = \deg P_\phi$, even arbitrarily many, if $\phi$ is chosen in an appropriate way.
2) If $R$ is not commutative, $R[X]$ can still be naturally endowed with the structure of a ring. However, evaluation is not necessarily a homomorphism anymore. In this case, are there normalised polynomials with an infinity of solutions? Matrix rings, perhaps? I am looking for a simple and intuitive example. Someone here who knows one?
So long, K.
