TITLE: What is the theory of polynomials?
QUESTION [16 upvotes]: We all know what polynomials are, along with their elementary properties and many effective algorithms for different representations of polynomials.
The question here is more of a universal algebra question: what is the signature of the theory which best corresponds to polynomials?  To illustrate what this question means, it is probably easiest to do this by example:

In the category of Unital Rings, the integers are the initial algebra.
In the category of semirings, the natural numbers are the initial algebra.

So what is a small presentation of a category (in the sense of giving signatures and axioms) for which the polynomials are initial?  Naturally, for $R[x]$, one expects that this presentation will either include the presentation of the ring $R$, or be parametric in that presentation.  But what else is needed to characterize univariate polynomial rings from general rings?
The motivation is that I am looking for a semantic type for univariate polynomials.  In most cases, the type for polynomials one encounters in the litterate is the type of its representations.  This is like saying that a matrix has semantic type 'square array', rather than to say that a matrix (in linear algebra) is a representation of a linear operator (with linear operator being the correct semantic type).

EDIT: one note of clarification.  After I figure out the 'theory of polynomials', I then wish to be able to write it down as well, so I want a 'presentation', universal-algebra-style, of the 'theory of polynomials' (whether that is plethories or free V-algebras on one generator or ...).  With the integers, it is easy to write down a set of axioms that define unital rings.

REPLY [3 votes]: IMHO the answer to “where polynomials are initial” (not to the title question, which is too broad for me) is already given in “Awodey. Category theory. 9. Adjoints. 9.3. Examples of adjoints. Example 9.10.”
In that example, the adjunction of functors is constructed, where its free (left adjoint) functor $F$ goes from the category of rings (=RingCat) to the category of rings with distinguished element (= pointed rings). If we define this adjunction via unit ($\eta$), then the definition says that

for every object $R$ in RingCat (= for
  every ring $R$) there is an
  initial object in the category $select(R)\downarrow U$ (comma
  category),

where $U$ is the forgetful functor.
Furthermore, a chosen initial object for $R$ consists of $F(R)$ (= $R[x]$) and $\eta(R):R\to U(F(R))$ (= a ring homomorphism constructing constant polynomials). The distinguished element in $F(R)$ is “$x$” (the projection polynomial).