Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces

We prove that a germ of a holomorphic map $f$ between $C^n$ and $C^{n^{\mathrm{\prime }}}$ sending one real-algebraic submanifold $M\subset C^n$ into another $M’\subset C^{n’}$isalgebraicprovided$M’$containsnocomplex-analyticdiscsand$M$ is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.