BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Tomáš Vávra (University of Waterloo) DTSTART:20201103T133000Z DTEND:20201103T143000Z DTSTAMP:20260423T040003Z UID:OWNS/22 DESCRIPTION:Title: Di stinct unit generated number fields and finiteness in number systems\n by Tomáš Vávra (University of Waterloo) as part of One World Numeration seminar\n\n\nAbstract\nA distinct unit generated field is a number field K such that every algebraic integer of the field is a sum of distinct unit s. In 2015\, Dombek\, Masáková\, and Ziegler studied totally complex qua rtic fields\, leaving 8 cases unresolved. Because in this case there is on ly one fundamental unit $u$\, their method involved the study of finitenes s in positional number systems with base u and digits arising from the roo ts of unity in $K$.\n \nFirst\, we consider a more general problem of posi tional representations with base beta with an arbitrary digit alphabet $D$ . We will show that it is decidable whether a given pair $(\\beta\, D)$ al lows eventually periodic or finite representations of elements of $O_K$.\n \nWe are then able to prove the conjecture that the 8 remaining cases ind eed are distinct unit generated.\n LOCATION:https://researchseminars.org/talk/OWNS/22/ END:VEVENT END:VCALENDAR