# Polynomials as concrete structures

### Motivation

The structuralist point of view on mathematical objects has two aspects:

On the one side, a mathematical object is seen as a concrete structure of abstract dots, e.g. a graph.

On the other side, a mathematical object is seen as an abstract dot in a concrete structure (of abstract dots), e.g. a category.

Most mathematical objects fit easily into both of these pictures: a graph, group, ring, field, etc. can be seen as a concrete structure of abstract dots, and as an abstract dot in a concrete structure, e.g. a category.

Even a single natural (= finite cardinal or ordinal) number fits into both pictures: as a specific dot in the infinite structure of natural numbers $\bullet\rightarrow\bullet\rightarrow\bullet\rightarrow\dots$, or as a bag of dots $\lbrace\bullet\bullet\dots\bullet \rbrace$, resp. a finite initial segment $\bullet\rightarrow\bullet\rightarrow\dots\rightarrow\bullet$ of the natural numbers.

The same goes e.g. for hereditary finite sets.

The natural numbers $\mathbb{N}$ as a whole do fit also: as an abstract dot in some category, and as the concrete structure $\bullet\rightarrow\bullet\rightarrow\bullet\rightarrow\dots$

### Question

For some objects I find it harder to fit them into both pictures. While it's easy to imagine an abstract ring that is isomorphic to a polynomial ring - with its dots representing "polynomials" - I have no idea what a polynomial is as a concrete structure (of abstract dots) in the sense above - or at least as a equivalence class of such concrete structures, equivalent with respect to the relation of "representing the same polynomial".

So:

Is there something like a polynomial as a concrete structure (of abstract dots)?

Or why is this question misled?

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A polynomial over a ring $A$ is a sequence $a_0,a_1, a_2,\dots$ of ring elements such that all but finitely many elements in the sequence ae $0$. So set-theoretically, it is a certain kind of function from the ordinal $\omega$ to $A$. Then one defines addition and multiplication as usual, as if the sequence $(a_n)$ were the polynomial we often call $a_0+a_1x+a_2x^2+\cdots$.

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A helpful way of seeing a polynomial is as an example of a monoid ring $A[M]$ where $A$ is the ring and $M$ is a monoid. The elements of $A[M]$ are written as finite sums

$$\Sigma_{i=1}^ra_i m_i$$

where $a_i \in A, m_i \in M$, and the product is given by

$$(\Sigma_i a_im_i)(\Sigma_j b_jn_j)= \Sigma_{i,j} (a_ib_j)(m_in_j).$$

The usual polynomials are when $M$ is the monoid of non negative integers under addition. This is the method that is adopted in the highly typed system AXIOM, which allows types as variables.

This reflects the method in modern mathematics of building more complicated structures by combination. Note that the type of a monoid ring is that of a ring. So you can use a monoid ring to form another monoid ring:

$$(A[M])[N],$$ and this is how we form polynomials in two variables.

As I understand it, the point of the AXIOM system is that a type specification specifies the operations that are allowed and the axioms these satisfy (but of course the axioms are not verified, but affect the representation of the objects).

So I am not sure how this fits with your notion of "dots". In mathematics, the dots themselves have a lot to them.

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thanks for your valuable answer! But what do you mean with your last sentence: "the dots themselves have a lot to them"? –  Hans Stricker Oct 1 '12 at 12:48
In mathematics one does a lot of treating a collection of things or a structure as an object, to describe its relation to other objects. But this is relevant to the notion of a name, part of the foundation of language, and so a foundation of our intellectual and social life. See the quotation from the bard on the role of the poet, on my popularisation page pages.bangor.ac.uk/~mas010/publar.html. Hope that helps. –  Ronnie Brown Oct 1 '12 at 19:41
Very interesting! I've thought a lot about names. And every now and then they seem to become important. –  Hans Stricker Oct 3 '12 at 14:20

If $R$ is a commutative ring, and $X$ is any singleton set, then $R[X]$ is the free commutative $R$-algebra of rank $1$. This generalizes easily to the case where $|X| = n$, where we get that the polynomial ring in $n$ variables is the free commutative $R$-algebra of rank $n$. If we drop the commutativity requirement, we get the free (unital, associative) $R$-algebra on $n$ generators.

This gives polynomials as left adjoints to forgetful functors. The "vector space" part of the $R$-algebra captures "polynomial addition" and "scaling", and the "ring part" of the $R$-algebra captures "polynomial multiplication" (including using the distributive law to "collect like terms").

I note in passing that this is equivalent to Ronnie Brown's answer, because the natural numbers are a free monoid over a singleton set (if you have a monoid, and you want to make it into a ring, you "polynomialize" (Is that a word? It should be!)).

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