I am taking a bit of time to study and understand deeply polynomials and their basic definitions and properties.
We define polynomials as expressions of the form $\sum_{i = 0}^n a_i x^i = a_0 + \ldots + a_n x^n $, where we take the sequence of coefficients $a_i$ to lie in some commutative (and probably unitary) ring $R$. In ring theory, is then defined the of set of polynomials in that ring:
$$R[x] := \Bigg \{ \sum_{i = 0}^n a_i x^i \mathrel{\bigg |} a : \mathbb{N}_{\leq n} \to R, n \in \mathbb{N} \Bigg \}$$
where we then define addition and multiplication, (I considered the coefficients as a sequence on the natural numbers $\{0,\ldots,n\}$) proving then that it is a ring. By the way, to make the usual theory and calculations, algebraists take the indeterminate "$x$" as a formal placeholder, a variable that in his essence is not made to be replaced with anything, differently from the function definitions $x \mapsto f(x)$, where $x$ will likely and soon be replaced. This makes the whole ring $R[x]$, in his nature, a set of informal expressions that are not functions nor any formal mathematical object, for which we define the behavior with the addition $+$ and the product $\cdot$.
In this set, two polynomials $\sum_{i = 0}^n a_i x^i$ and $\sum_{i = 0}^n b_i x^i$ are defined (axiomatized, in some sense) to be equal if and only if their coefficients $a_i$ and $b_i$ are, for all $i \in \mathbb{N}_{\leq n}$, that is, if and only if $a = b$ as sequences.
Now, to every of such polynomial expression we can associate a polynomial function, which now is a function of expression
$$f(x) := \sum_{i = 0}^n a_i x^i $$
But now, as polynomial functions, how do we prove that the functions $f(x) = \sum_{i = 0}^n a_i x^i $ and $g(x) = \sum_{i = 0}^n b_i x^i $ are equal if and only if $a=b$, that is, the sequences of coefficients are coincident $\large ?$ I am not sure, but maybe for polynomials $\mathbb{C} \to \mathbb{C}$ over the ring $\mathbb{C}$ the fundamental theorem of algebra could be useful...$\large ?$
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Going on a bit higher and general level, the operations which are used in a polynomial expression $\sum_{i=0}^n a_i x^i$ are the exponentiation of $x \in D$ (I'll call its lying set D), the multiplication of $a_i \in R$ with $x$ and finally the addition of the obtained terms. Thus, we need a scalar multiplication $\cdot_R : R \times D \to D$, an addition $+: D \times D \to D$ and an exponentiation $-^- : \mathbb{N} \times D \to D$, which potentially comes from a multiplication $\cdot : D \times D \to D$. In a way of thinking about it, we'd need the exponentiation $x^0$ to be equal to some kind of multiplicative identity $1_D$. Summing all these conditions, we could assume that $D$ itself is a commutative unitary ring together with a "scalar" multiplication $\cdot_R : R \times D \to D$, or maybe we could assume $D$ to be some kind of "$R$-algebra" (being a module it has automatically a scalar multiplication) with an internal multiplication $D \times D \to D$ that has an identity $1_D$. These list of possible assumptions on the lying set $D$ of $x$ bring me to the question: how can we sum up the minimal structural conditions of $D$ in order to being able to consider polynomial functions $D \to D$ with coefficients in a ring $R$ $\large ?$
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In this way, I thought that it's actually possible to give a (bit complicated) formal defintion of what really a general polynomial function is: whatever structure $D$ we have to consider on a ring $R$, -I'll summarize this property of $D$ writing $D \in \textit{str}(R)$-, we can think of a general polynomial with coefficients in $R$ as a family of polynomial functions on the collection of structures $\textit{str}(R)$:
$$ P \in \prod_{D \in \textit{str}(R)} D \to D \hspace{30pt} P_D(x) := \sum_{i = 0}^n a_i x^i$$
and being inside the set theoretic product $\prod_{D \in \textit{str}(R)} D \to D$ makes $P$ a family that can be then observed by the point of view of any $D$, so that $P_D : D \to D$ for any such structure.
We can express the same object $P$ as a family of polynomial functions in the product $\prod_{D \in str(R)} D \to D$ also using the $\lambda$-calculus formalism, in which, bounding $x \in A$, $\lambda x \in A . f(x) : A \to B$ represents the unique function $h : A \to B$ such that $h(x) = f(x)$. With this compact notation, the "new" ring of general polynomials would be
$$R[x] := \Bigg \{ \lambda D \in \textit{str}(R).\lambda x \in D. \sum_{i = 0}^n a_i x^i \mathrel{\bigg |} a : \mathbb{N}_{\leq n} \to R, n \in \mathbb{N} \Bigg \}$$
being now a set of specified and pre-existing mathematical objects. In this new $R[x]$, as in the old one, we should still, after having defined correctly addition and multiplication on it, get a ring isomorphism $R[x] \cong (\mathbb{N} \xrightarrow{\text{ ev 0}} R)$ with the sequences of coefficients of elements of $R$ that go eventually to $0_R$. In proving this relation, anyway, (actually just the bijection between the two) we come back to the need of proving that if two general polynomials functions $P$ and $Q$ are equal -as functions-, then their sequence of coefficients coincide. Is there a way to do this with this new definition $\large ?$ Is there a meaningful minimal structure $str(R)$ for $D$ which makes us able to prove it (maybe considering $D$ as a field) or which has a "fundamental theorem" that can help $\large ?$
