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+--- Thread: " fake ring theory " (/showthread.php?tid=885)
" 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.
regards
tommy1729