### - and ${}^{\infty}$-hypoellipticity of partial differential operators with constant Colombeau coefficients

We provide a deep investigation of the notions of - and ${}^{\infty}$-hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and ${}^{\infty}$-hypoellipticity are given