We all know [what polynomials are](http://en.wikipedia.org/wiki/Polynomial), 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).
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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.