Global hypoellipticity for a class of overdetermined systems of pseudo-differential operators on the torus

This article studies the global hypoellipticity of a class overdetermined systems pseudo-differential operators defined on torus. The main goal consists in establishing connections between system and its normal form. It is proved that an obstruction number-theoretical nature appears as necessary condition to hypoellipticity. Conversely, sufficiency approached by analyzing three types hypotheses: Hörmander condition, logarithmic growth super-logarithmic growth.