Tejinder Neelon
Restrictions of power series and functions to algebraic curves
The analogs of the following theorems in more general set up of classes of smooth functions and subrings formal power series will be presented.
(i) (Hartogs) A function of several complex variables is analytic if it is analytic separately in each variable.
(ii) (Lelong) A formal power series in $n$ variables that converges on a nonpolar set of lines through the origin is necessarily convergent.
(iii) (Bochnak-Siciak) If an infinitely differentiable function $f$ is real-analytic on every line segment through a point $p$ then $f$ is real-analytic in a neighborhood of $p$.