Homogeneous sliding mode (SM) control is known to provide for finite-time-exact output regulation of uncertain systems with known relative degrees. Yet the corresponding universal homogeneous SM (HSM) controllers are typically constructed by special recursive procedures and have complicated form. New HSM controllers are based on a very simple polynomial homogeneous form. Lyapunov functions are provided for a significant part of controllers. Quasi-continuous versions of these controllers can be done arbitrarily smooth everywhere outside of the corresponding SM.