Patrick Speissegger
In this appendix to the Fields 2022 module, I briefly discuss a couple of loose ends left in this post. First, let $f = (f_0, \dots, f_k) \in \U^{k+1}$ be such that $f_0 > \cdots > f_k$. We saw that if $f_0 \succ \cdots \succ f_k$, then our construction yields a qaa field $(K_f, e^x…
(Joint work with Zeinab Galal and Tobias Kaiser) I describe here the generalization of this construction to arbitrary convergent transmonomials, based on the Continuation Theorem. Definition A germ $f \in \I$ is simple if $\level(f) = \eh(f)$. For example, every convergent transmonomial is simple, as is every purely infinite germ. Recall that $$\la := \bigcup_{n…
(Joint work with Tobias Kaiser) I introduce real domains in $\LL$ and angular level, and I use these notions to describe holomorphic continuations on $\LL$ of one-variable functions definable in $\Ranexp$. In this post “definable” means “definable in $\Ranexp$”. DefinitionA set $U \subseteq \LL$ is a real domain if there exist $a \gt 0$ and…
(Joint work with Tobias Kaiser; this is a repost) We are interested in holomorphic continuations of one-variable functions definable in $\Ranexp$. Since $\exp$ and $\log$ are two crucial functions definable in $\Ranexp$, the natural domain on which to consider holomorphic continuations of all definable functions is the Riemann surface of the logarithm $$\LL:= (0,\infty) \times…
(Joint work with Tobias Kaiser; this is essentially a reposting of this post) The goal of this post is to describe the Hardy field $\H = \Hanexp$ of the expansion $\Ranexp$ of the real field by all restricted analytic functions and the exponential function, based on van den Dries, Macintyre and Marker’s papers on $\Ranexp$…