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.



What is the theory of polynomials?
My first reaction to this is that you might be interested in the general theory of [Tall-Wraith monoids](http://ncatlab.org/nlab/show/Tall-Wraith+monoid). The special case which might be of more immediate interest to you is the notion of **plethory**. A good reference is [this paper](http://maths.anu.edu.au/~borger/papers/03/lambda.pdf) by Borger and Wieland (you will see your polynomial algebra on page 2). 


It works like this: suppose you have a (commutative and cocommutative) [biring](http://ncatlab.org/nlab/show/biring) $H$ over a commutative ring $R$. Just having a commutative algebra $A$ over $R$ gives you a representable functor 


$$\hom(A, -): CAlg\_R \to Set$$ 


but having a biring $H$ over $R$ means you have enough structure to lift the representable to a functor 


$$\hom(H, -): CAlg\_R \to CAlg\_R$$ 


Such endofunctors on $CAlg\_R$ can be composed, and it turns out that there is a monoidal product $\odot$ on birings which represents this composition, in the sense of there being a canonical isomorphism 


$$\hom(H \odot H', -) = \hom(H, -) \circ \hom(H', -)$$ 


A **plethory** is by definition a monoid object in this monoidal category of birings. 


Getting to your question, the biring which represents the identity functor on $CAlg\_R$ is the polynomial algebra $R[x]$. This is therefore the monoidal unit in the monoidal category of birings, or the initial object in the category of plethories over $R$. 


Therefore it seems you want to study the universal algebra of plethories. This is probably best carried out in context of [props](http://ncatlab.org/nlab/show/PROP) whose semantics is valued in a symmetric monoidal category, particularly the symmetric monoidal category $Mod\_R$. 


A local MO expert in these matters is Andrew Stacey. 


