I have seen many questions addressing the question to how many integer solutions exist to the following equation: \begin{equation} y_1 s+y_2+...+y_n=i, \end{equation} with each of the $n$ variables $\{y_1,...,y_n\}$ belonging to the $same$ set, being it infinite or finite. The problem in those case boils down to computing the coefficients of the related generating function, as in this exhaustive question, which considers the diffrent cases of $y_i$ being $\ge 0$ or $y_i$ belonging to the finite set $[d,u]$, with $d<u$ integers.
I am curious to see what happens when the $n$ variables $\{y_1,...,y_n\}$ belongs to $different$ sets. I propose the following.
Let's consider the following first order polynomial equation in $n$ variables $\{x_1,...,x_n\}$, where each $x_i$ is within the finite set $\{0,1,...,F-1,F\}$ made by all integers between $0$ and $F$: \begin{equation} x_1 s+x_2s^2+...+x_ns^n=i, \end{equation} where $s$ and $i$ are whatever positive integers.
My goal is to get the number of solutions $\{x_1,...,x_n\}$ to this polynomial equation, given $i$.
I remark that if $F=(s-1)$ I am simply looking at how many ways I can rewrite $i$ in a $s$-basis, hence there would be just 1 combination of $\{x_1,...,x_n\}$ allowing that.
