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" fake ring theory " - tommy1729 - 06/11/2014

Ok so this might be a strange title.

Its the idea to combine fake function theory with ring theory.

I discussed this today with my friend mick.

Consider reduced real polynomial rings like F(X) := R[X]/(G(X)).

Now we can define the exp for every element z of that ring by simply using the Taylor series of exp(z) mod(G(X)).

Since exp(z) is entire that works fine.

Now come the key ideas :

exp(X) mod (G(X)) must be a polynomial in X !
Call that P(X).

Now from fake function theory : exp^[1/2] is also " entire ".

And then " fake function theory " gives the approximate equality :

exp^[1/2](X) ~ P^[1/2](X).

Depending on the degree of G , the selection of z ( Here it was z = X ) , the type of ring etc etc, we can go many directions with this idea.

Multisection theory for instance.

Asymptotic analysis , Galois theory , ...

And it can even be used for other function than exp.

Now ring elements can be linked to complex numbers , real numbers , matrices etc.

To end with style :

A > 0
exp^[A](X) ~ P^[A](X).

Notice many rings do not have log so A > 0 is necc.