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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. regards tommy1729 |