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.