Let $ p ( x ) \in \mathbb N [ x ] $ be a polynomial with nonnegative integer coefficients, and $ a \in \mathbb Z $ be a given integer constant. If for all positive integers $ n $, $ p ( x ) + a $ has a unique root in the finite ring $ \mathbb Z _ n $, can we conclude that $ p ( x ) $ must be linear?
This problem comes from another one that I'm working on, and can be considered as an special case weaker than the original problem. While working, I realized that I have no method for solving this special case either, and I need to become sure that it holds, first.
The intuition behind why I think $ p ( x ) $ must be linear comes from Schur's conjecture (it is now a theorem in fact, yet still widely referred to by this name). If instead of having a unique root, $ p ( x ) + a $ were supposed to be a permutation polynomial, then by the conjecture it should have been the composition of linear functions and Dickson polynomials (of the first kind). But since all the coefficients of $ p ( x ) $ are nonnegative, and Dickson polynomials have a certain pattern of alternating signs in coefficients, it seems that $ p ( x ) $ must be linear (I haven't checked this in full detail, but it seems to be true). That made me more confident in believing that the nonnegativity of the coefficients of $ p ( x ) $ might be a strong enough condition to add to the weaker assumption of $ p ( x ) + a $ having a unique root, to get the same result. Also note that while the assumption of Schur's conjecture is that the polynomial is a permutation polynomial in the finite field $ \mathbb Z _ p $ for infinitely many prime numbers $ p $, here we have the unique root property in $ \mathbb Z _ n $ for all positive integers $ n $, which seems to be a very strong restriction.
The above mentioned theory of permutation polynomials was the closest theory to my problem in the literature that I could find searching online. I couldn't find much information on polynomials having unique roots when considered over several finite fields or rings. This may come from my unfamiliarity with the subject, of course. I find it very helpful if you could give references to parts of the literature more directly related to the problem at hand.
The positivity assumptions are irrelevant. So more generally, suppose $q\in\mathbb{Z}[x]$ has exactly one root mod $n$ for all $n$. Let $f$ be any irreducible factor of $q$. By the Chebotarev density theorem, there are infinitely many primes $p$ such that $f$ splits as a product of distinct linear factors mod $p$. By our assumption on $q$, this means $f$ must be linear.
So, $q$ must actually be a product of linear factors over $\mathbb{Z}$, i.e. all its roots are rational. If $q$ has two distinct rational roots, there is some $p$ such that those roots exist and remain distinct mod $p$ (just take $p$ that does not divide their denominators or the numerator of their difference). So, $q$ can only have one rational root, so $q(x)=a(bx+c)^n$ for some $a,b,c\in\mathbb{Z}$ and some $n\in\mathbb{N}$ with $b$ and $c$ relatively prime. Clearly $a$ must be $\pm 1$ and then $b$ must also be $\pm 1$ (otherwise there would be no root mod $b$). Replacing $x$ with $bx+c$ (which is an invertible change of variables), we may thus assume $q(x)=x^n$. But now if $n>1$, $q$ has more than one root mod $p^n$ for any prime $p$ (namely both $x=0$ and $x=p$ are roots), so $n$ can only be $1$.