BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Antoine Ducros (Sorbonne Université) DTSTART:20230505T144500Z DTEND:20230505T154500Z DTSTAMP:20260422T172402Z UID:TGiZ/52 DESCRIPTION:Title: Tr opical functions on skeletons: a finiteness result\nby Antoine Ducros (Sorbonne Université) as part of Tropical Geometry in Frankfurt/Zoom TGiF /Z\n\n\nAbstract\nSkeletons are subsets of non-archimedean spaces (in the sense of Berkovich) that inherit from the ambiant space a natural PL (piec ewise-linear) structure\, and if $S$ is such a skeleton\, for every invert ible holomorphic function $f$ defined in a neighborhood of $S$\, the restr iction of $\\log |f|$ to $S$ is PL. \nIn this talk\, I will present a join t work with E.Hrushovski F. Loeser and J. Ye in which we consider an irred ucible algebraic variety $X$ over an algebraically closed\, non-trivially valued and complete non-archimedean field $k$\, and a skeleton $S$ of the analytification of $X$ defined using only algebraic functions\, and consis ting of Zariski-generic points. If $f$ is a non-zero rational function on $X$ then $\\log |f|$ induces a PL function on $S$\, and if we denote by $E $ the group of all\nPL functions on $S$ that are of this form\, we prove t he following finiteness result on the group $E$: it is stable under min an d max\, and there exist finitely many non-zero rational functions $f_1\,\\ ldots\,f_m$ on $X$ such that $E$ is generated\, as a group\nequipped with min and max operators\, by the $\\log |f_i|$ and the constants $|a|$ for $ a$ in $k^*$. Our proof makes a crucial use of Hrushovski-Loeser’s model- theoretic approach of Berkovich spaces.\n LOCATION:https://researchseminars.org/talk/TGiZ/52/ END:VEVENT END:VCALENDAR