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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:

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    @Asaf. Added some refernces. You can also try the thesis by Joris van der Hoeven. Unfornately, this is in French and my French isn't that good...2012-07-25

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