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.

As a side note, the "form-function epimorphism", and the structure defined by different collections of these morphisms (arising from different indeterminates and different rings) appears to be quite an interesting one. As far as I can tell, there is little literature on the subject; probably because at first sight the relationship between forms and functions appears rather more trivial than I'm beginning to think it is.

For polynomial forms
In this case the "induced structure lemma" has to be applied to two polynomials which according to the definition are maps $M\to (R,+,\circ)$ from the Definition:Free Commutative Monoid of mononomials to a commutative ring with unity. So for a Definition:Mononomial $\mathbf X^k$, let


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


 * $g:\mathbf X^k\mapsto g_k$

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 evaluation at a 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
In this case the polynomials are maps $R^J\to R$, so we are applying the induced structure definition to a different object, the symbol $f(x)$ now means "evaluate the polynomial at $x\in R^J$" not "project to the coefficient of the mononomial $x$". Let


 * $\displaystyle f=\sum_k a_k\mathbf X^k$


 * $\displaystyle g=\sum_k b_k\mathbf X^k$

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 poly. functions is $(f\circ g)(x)=\sum_k\sum_{p+q=k}a_pb_q\mathbf X^k$. Evaluating this gives, for $s\in R^J$,


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

For the induced-structure definition, we evaluate first


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

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 the whole polynomial, some of which is lost when we evaluate at a particular $X^k\in M$.

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


 * Multiplication continues to work because information about every coefficient is retained.