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SUMMARY:Gwladys Fernandes (Université de Versailles Saint-Quentin-en-Yvel
 ines)
DTSTART:20220509T110000Z
DTEND:20220509T120000Z
DTSTAMP:20260423T022933Z
UID:WarsawNT/42
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WarsawNT/42/
 ">Hypertranscendence of solutions of linear difference equations</a>\nby G
 wladys Fernandes (Université de Versailles Saint-Quentin-en-Yvelines) as 
 part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\
 n\nAbstract\nThe general question of this talk is the classification of di
 fferentially algebraic solutions of linear difference equations of the fol
 lowing type:  \n\n$$ (\\ast) \\qquad a_0(z)f(z) + a_1(z)f(R(z)) + ... + a_
 n(z)f(R_n(z))=0$$\n\nwhere\, for every $i$\, $a_i(z) \\in \\C(z)$\, $R(z) 
 \\in \\C(z)$ and $R_n(z)$ is the $n$-th composition of $R(z)$ with itself.
  We say that such a function is differentially algebraic over $\\C(z)$ if 
 there exist an non-zero integer $n$ and a non-zero polynomial $P \\in \\C(
 z)[X_0\,...\, X_n]$ such that $P(f(z)\,...\, f^{(n)}(z))=0$\, where $f^{(i
 )}$ is the $i$-th derivative of $f$ with respect to $z$. Otherwise\, it is
  hypertranscendental over $\\C(z)$.\n\nThe classification of differentiall
 y algebraic solutions is known for three types of non-linear difference eq
 uations : the Schröder's\, Böttcher's and Abel's equations : $f(qz)=R(f(
 z))$\, $f(z^d)=R(f(z))$\, $f(R(z))=f(z)+1$\, respectively\, where $q \\in 
 \\C^{\\ast}$\, $d \\in \n$\, $d \\geq 2$. A classification of the differen
 tial algebraicity of solutions of linear difference equations of the above
  type $(\\ast)$ is made in an article of B. Adamczewski\, T. Dreyfus\, C. 
 Hardouin\, for these same operators : q-differences $z \\to qz$\, mahleria
 n $z \\to  z^d$\, and shift $z \\to z+1$\, by the means of an adapted diff
 erence Galois theory.\n\nIn this talk\, we discuss the generalisation of t
 hese results to any function $R$ (rational or algebraic over $\\C(z)$)\, i
 n the case where $(\\ast)$ is of order $1$. This is a work in progress wit
 h L. Di Vizio. Natural applications appear in examples of generating serie
 s of random walks\, which satisfy this kind of equation of order $1$.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/42/
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