# Definition talk:Pointwise Operation

## Polynomials

### To begin

Phew! This turned out to be quite an essay, I only sat down to write a few sentances...

- There may be a way to make the induced definition work for polynomial forms too, but I can't think of it

- I hope the following makes enough sense to be worth reading.

The question arose because I wanted to elucidate the distinction between polynomial forms and polynomial functions via an epimorphism from the former to the latter; this means that polynomials forms have a precise foundation without interfering with all the existing notation used relating to polynomials considered as functions.

### For polynomial forms

- By definition, a polynomial form is a map $M\to R$ from the free commutative monoid of monomials to R.

- For a monomial $\mathbf X^k \in M$, the symbol $f(\mathbf X^k)=a^k$ means "project $f$ to the coefficient of the monomial of $\mathbf X^k$".

Let

- $f:\mathbf X^k\mapsto a_k$

- $g:\mathbf X^k\mapsto b_k$

be polynomial forms.

#### Addition

Polynomial addition is defined by $(f+g)(\mathbf X^k)=a_k+b_k=f(\mathbf X^k)+(\mathbf X^k)$, which corresponds exactly with the "induced structure definition".

#### Multiplication

Here, $\displaystyle (f\circ g)(\mathbf X^k) = \sum_{p+q=k}a_pb_q$. In this case, in general $(f\circ g)(\mathbf X^k)\neq f(\mathbf X^k)\circ g(\mathbf X^k)=a_kb_k$.

That is, multiplication of polynomials does not commute with projection to the coefficient particular monomial.

By the induced structure definition of multiplication this *has* to be the case, because the definition says that the two must commute. In particular the induced definition is

- $(f\circ g)(\mathbf X^k)=f(\mathbf X^k)\circ g(\mathbf X^k)$,

which is not the needed operation.

### For polynomial functions

- By definition, a polynomial function is a map $R^J\to R$ from the free module on $J$ to $R$

- For a `point' $x \in R^J$, the symbol $f(x)$ means "evaluate $f$ at the point $x$".

Let

- $f(x)=\sum_{k\in Z}a_kx^k$

- $g(x)=\sum_{k\in Z}b_kx^k$

be polynomial functions.

#### Addition

Addition is $(f+g)(x)=f(x)+g(x)$, the same as the induced structure gives.

#### Multiplication

Now it is different:

The definition (the non-induced structure definition) of multiplication of polynomial functions is $(f\circ g)(x)=\sum_k\sum_{p+q=k}a_pb_qx^k$. Evaluating this gives, for $s\in R^J$,

- $(f\circ g)(x)=\sum_k\sum_{p+q=k}a_pb_q x^k$

For the induced-structure definition, we evaluate first

- $f(s)\circ g(s)=\left(\sum_k a_k x^k\right)\left(\sum_k b_k x^k\right)=(f\circ g)(x)$.

So in this case, morally speaking "multiply then evaluate equals evaluate then multiply".

### Cause of the difference

In the case of polynomial forms, we have to consider the map we use for the induced structure definition to be "projection to the coefficient of a particular monomial".

When we apply this map we lose the information about the remaining coefficients, which are needed to evaluate the product.

Therefore, for polynomials forms, multiplication and evaluation (at a monomial) do not commute. For forms:

- Addition works because additive information is local to a monomial, and all non-local information is lost under the map $f:M\to R$

- Multiplication does not work because multiplicative information requires information about all the coefficients of the polynomials, some of which is lost when we project to the coefficient of some $X^k\in M$.

In the case of functions, "evaluation" has a different meaning; it retains information about *all* the coefficients of a polynomial. Loosely speaking, $f:R^J\to R$ does not `forget' global information. Therefore we can use the induced structure to define mulitplication also. For functions:

- Multiplication works because information about every coefficient is retained.

## Merge with Definition:Pointwise Addition of Mappings

I agree with the merge, of course; though it appears that it is generally hard to determine whether a certain concept has already been covered on PW. This is not completely separated from me using Google with 'site:proofwiki.org' instead of the internal search, which I deem quite inferior. This concludes in a plead for more redirects and alternative nomenclature statements. --Lord_Farin 08:02, 24 January 2012 (EST)

Additional comment: I feel the page could do with more explicit naming, like 'Induced Structure on Set of Mappings'. Nobody will find this page when looking for what pointwise addition, multiplication etc. are. --Lord_Farin 08:06, 24 January 2012 (EST)

The statement that it is again an algebraic structure needs proof; it might be desirable to cast the proof in the language of category theory, disposing of any unnecessary details of the specific structure; but then, the foundations of category theory still lie in darkness. I recently bought MacLane's authoritative 'Categories for the Working Mathematician', so when time comes, this will be covered. --Lord_Farin 06:07, 15 March 2012 (EDT)