Definition talk:Pointwise Operation

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

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

For polynomial forms

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


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

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 mononomial".

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 mononomial) do not commute. For forms:


 * Addition works because additive information is local to a mononomial, 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.