# Formal intuition behind a polynomial as function

Let $$R$$ be a ring, then define the polynomial space $$R[x]$$ as

$$R[x]=\{ (a_0, a_1, \ldots) : a_j \neq 0 \text{ for a finite number of } a_j \in R \}.$$

So formally, a polynomial is just a tuple. But how can I imagine it as function? What is $$f(c)$$ formally? It's clear that $$f(c)$$ is just the value of the polynomial applied on $$c$$ but how does this harmonize with the definition of $$f$$ not as function but as tuple of coefficients?

And what is $$x$$ in $$R[x]$$? It doesn't appear in the definition so what is the difference between $$R[x]$$ and $$R[y]$$? Why does one specifies the polynomial space on $$x$$?

Of course, a tuple is not a function. Formally, you have a natural morphism $$\ell: R[x] \rightarrow \{f:R \rightarrow R\}$$ given by $$\sum_n{a_nx^n} \longmapsto (c \longmapsto \sum_n{a_nc^n})$$.
So, if $$f$$ is a polynomial and $$c \in R$$, then what is denoted as $$f(c)$$ is actually $$(\ell(f))(c)$$.
In $$R[x]$$, $$x$$ is formally how you name the sequence $$(0,1,0,\ldots)$$. You specify it because it’s nicer to write expressions as sums of powers of $$x$$ than as stationary sequences.
Moreover, there are polynomials in several variables, and it’s then nice to be able to differentiate between several variables. It is also useful to specify $$R[x]$$ when $$R$$ is already a ring of polynomials (or, eg, formal series), with respect to another variable, so that you don’t mix them up.
• But why is - notation wise - $\sum_n a_n x^n \in R[x]$? How and especially why can I interpret a tuple as sum?
• Actually, what happens is that you can add and multiply sequences between themselves (addition is pointwise, product follows the rule for polynomials) and multiply a given sequence by an element of $R$ (pointwise). So, for each $n$, $a_nx^n$, in the definition of my post for $x$, is the sequence that is $0$ everywhere outside $n$ and is $a_n$ at $n$. So $\sum_n{a_nx^n}$ is the sequence that is at any $n$, $a_n$. May 2, 2020 at 15:21