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SUMMARY:Charlotte Hardouin (Institut de mathematiques de Toulouse)
DTSTART:20201113T170000Z
DTEND:20201113T175000Z
DTSTAMP:20260422T221129Z
UID:20w5206/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/20w5206/11/"
 >Algebraic independence of solutions of linear difference equations</a>\nb
 y Charlotte Hardouin (Institut de mathematiques de Toulouse) as part of BI
 RS workshop: Algebraic Dynamics and its Connections to Difference and Diff
 erential Equations\n\n\nAbstract\nThis work is a collaboration with B. Ada
 mczewski (ICJ\, France)\, T. Dreyfus (IRMA\, France) and M. Wibmer (Graz U
 niversity of Technology\, Austria). \n\nIn this talk\, we will consider pa
 irs of automorphisms $(\\phi\,\\sigma)$ acting on fields of Laurent or Pui
 seux series:  pairs of shift operators\, of $q$-difference operators  and 
 of Mahler operators.  Assuming that the operators $\\phi$ and $\\sigma$ ar
 e "independent"\, we show that their solutions are also "independent" in t
 he sense that a solution $f$ to a linear $\\phi$-equation and a solution $
 g$ to a linear $\\sigma$-equation are algebraically independent over the f
 ield of rational functions unless one of them is a rational function.  As 
 a consequence\, we settle a conjecture about Mahler functions put forward 
 by Loxton and van der Poorten in 1987. We also give an application  to the
  algebraic independence of  $q$-hypergeometric functions.  \n Our approach
  provides a general strategy to study this kind of questions and is based 
 on a suitable Galois theory: the $\\sigma$-Galois theory of linear $\\phi$
 -equations developed by Ovchinnikov and Wibmer.\n
LOCATION:https://researchseminars.org/talk/20w5206/11/
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