$n,m\in \mathbb{N}, m\geq2$ determine the number of polynomials with coefficients from the set $\{0,1...,m^2-1\}$ that satisfy $P(m)=n$ 
$n,m\in \mathbb{N}, m\geq2$ determine the number of polynomials with coefficients from the set $\{0,1...,m^2-1\}$ that satisfy $P(m)=n$

My attempt:
I'm really not sure how to approach this problem. I'm not even sure if $m$ and $n$ are fixed numbers? Because if not the obvious solution would be $1$, $P(x) = n$. I would attempt something more but I'm really not sure how to go about this.
I haven't really seen any similar problems so I'm not sure how to approach this.
I'm asking for recreation, just looking for some hints. Thanks a lot!
 A: Hint: Find $k_1$ and $k_2$ such that:
$m^{k_2} > n$ and $ \sum_{i=0}^{k_1} (m^2-1) m^{i} <n$. Then any polynomial P must have degree between $k_1+1$ and $k_2-1$
A: $m$ and $n$ are fixed numbers. You are asked to find the total number of such polynomials; this will depend on $m$ and $n.$ The degree-zero polynomial $n$ will be one such polynomial if and only if $n\leq m^2-1.$ But for example if $n=3$ and $m=2,$ you also need to count the polynomial $x+1$ as well as the polynomial $3.$ (Are there others?)
The first question that comes to mind is: are there a finite number of such polynomials? You should be able to show there are.
It might help to consider the easier problem of counting polynomials $P$ with coefficients in $\{0,\dots,m-1\}$ such that $P(m)=n.$
A: I would look to a recursive computation and will describe how it would work for $m=3,n=1000$.  The first thing is to establish the highest degree of the polynomial.  Since $3^6=729, 3^7=2187$ we cannot have any polynomials of seventh degree or higher.  If the polynomial is sixth degree, the leading term must be $1x^6$ because $2x^6$ is already too large.  If the leading term is $x^6$ you are now looking for polynomials of degree at most $5$ with $P(3)=271$.  If there is no $x^6$ you are now looking for polynomials of degree at most $5$ with $P(3)=1000$.  In the second case the largest coefficient of the $x^5$ term is $4$, so look for fourth degree polynomials with $P(3)=1000, 757, 514, 271, 28$ and so on.  As the coefficients cannot get greater than $8$ in our example you will find cases where you can't get the total high enough.  The case with no $x^5$ or $x^6$ terms is one.  The greatest fourth degree polynomial is $8x^4+8x^3+8x^2+8x+8$ and at $x=3$ this is $968$
