BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Sylvy Anscombe (Université Paris Cité) DTSTART:20240612T150000Z DTEND:20240612T160000Z DTSTAMP:20260418T065733Z UID:LNTS/134 DESCRIPTION:Title: U niform aspects of the theory of complete valued fields\nby Sylvy Ansco mbe (Université Paris Cité) as part of London number theory seminar\n\nL ecture held in K0.18\, King's Building\, Strand Campus\, King's College Lo ndon.\n\nAbstract\nA good deal of the arithmetic of a field can be express ed by sentences in the first-order language of rings. The theories\nof the characteristic zero local fields have been axiomatized and are decidable: in the case of $Q_p$ and its finite extensions\,\nAx\, Kochen\, and (inde pendently) Ershov\, gave complete axiomatizations that are centred on a fo rmalization of Hensel’s\nLemma. In fact the theory of any field of chara cteristic zero which is complete with respect to a non-archimedean\nvaluat ion can be likewise axiomatized.\n\nI will explain recent joint work with Jahnke\, and also with Dittmann and Jahnke\, in which we extend the classi cal\nwork on these theories to include the case of imperfect residue field s. In particular we show that “Hilbert’s Tenth\nProblem” (H10) in th ese fields (i.e. the problem of effectively determining whether a given Di ophantine equation has\nsolutions) is solvable if and only if the analogou s problem is solvable on a structure we define on the residue field. This\ nfollows a pattern of such “transfer” results for H10 — established for valued fields of positive characteristic in earlier\nwork with Fehm — although in the current case we really need the extra structure.\n\nI will describe these results\, focusing on the extent to which they depend (or not) on the residue field. If there is\ntime I will discuss the aforem entioned H10 transfer for complete valued fields in positive characteristi c\, including more\nrecent uniform aspects.\n\nI will not assume a backgro und in logic.\n LOCATION:https://researchseminars.org/talk/LNTS/134/ END:VEVENT END:VCALENDAR