Departmental Colloquium

Alex Wilkie
Remarks on the zero sets of exponential polynomials
Abstract: Please note the room for this talk is 613 Student Center East.
Let $K$ be a subfield of the complex field $\mathbb{C}$. By an exponential polynomial over $K$ we mean a function of the form $P(z_0, \ldots , z_n , e^{z_0}, \ldots , e^{z_n})$ where $P$ is a polynomial over $K$. In this talk I discuss complex space curves (in $\mathbb{C}^{n+1}$) given as the intersection (assumed everywhere nonsingular with respect to the variables $z_1, \ldots , z_n$) of the zero sets of $n$ such exponential polynomials. Let $\Omega$ be such a curve and let $\pi_0 [\Omega]$ be its projection on to the $z_0$-plane. Then $\pi_0 [\Omega]$ is an open subset of $\mathbb{C}$ and one expects it to be co-countable. Even this special case of Zilber's famous conjecture on the complex exponential field (which will be explained in the talk) is, as far as I know, still unknown. But I shall present a result that implies that the open set $U \cap \pi_0 [\Omega]$ is dense (in $U$) and connected for every connected open subset $U$ of $\mathbb{C}$. This has the model theoretic consequence that the set of reals does not lie in the $\sigma$-algebra generated by the subsets of $\mathbb{C}$ defined in the complex exponential field by existential formulas.
Please note the room for this talk is 613 Student Center East.
Friday October 26, 2018 at 3:00 PM in 613 SCE
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