On polynomial solutions of linear partial differential and (q-)difference equations

par Sergei A. Abramov (Computing Centre of the Russian Academy of Sciences)

jeudi 1 déc. 2011, 10:30 → 11:30 Europe/Paris

XR.203 (Bâtiment XLIM)

The question whether a given linear partial differential, difference or $q$-difference equation with polynomial coefficients has non-zero polynomial solutions or not, is -- in general -- undecidable. However, if a differential or difference equation $L(y)=0$, $y = y(x_1,\ldots,x_m)$, $m>1$, with constant coefficients has a non-zero polynomial solution then $L(1)=0$, and if $L(1)=0$ then the equation has polynomial solutions of all degrees. For a given non-negative integer $d$, all solutions of degree $d$ of such an equation can be found by, e.g., the method of undetermined coefficients. The space of polynomial solutions of a given $q$-difference equation with constant coefficients can be described algorithmically, and this space may be of finite or infinite dimension. The results presented in this talk were obtained by the author jointly with M.Petkovsek.