In the quest for extensions of $\mathbb{R}$ and $\mathbb{C}$ that contains infinitesimals, infinities (and even more exotic beasts like $\omega - 1$ and $\sqrt{\omega}$) I came across the theory of transseries, for which there is a claim that it could be isomorphic to the surreals. This is not proven yet, but I have a few questions:
- Is $\{P(z) \in \mathbb{C}[z^a,e^{L_\alpha(z)}]$ with $a \in \mathbb{C}$ and $L(z) \in \mathbb{C}[z^a,e^{L_\beta(z)}]$ a correct description, given you "sort" out the recurrent definition here, of (log-free) transseries. I am aiming here for a more algebraic "plane and simple" definition. Yes, I know this is kinda vague.
- What are examples of elements of the field of transseries that are infinitesimals ?
- What are examples of elements of the field of transseries that are infinite elements ?
I don't know whether the number theory tag applies here. I invite you to put something better here.
My References are:
- http://arxiv.org/abs/0801.4877 (transseries for beginners)
- http://www.texmacs.org/joris/ln/ln-abs.html (book by Joris van der Hoeven)