  We study the effects of cut-off physics, in the form of a modified algebra
inspired by Polymer Quantum Mechanics and by the Generalized Uncertainty
Principle representation, on the collapse of a spherical dust cloud. We analyze
both the Newtonian formulation, originally developed by Hunter, and the general
relativistic formulation, that is the Oppenheimer-Snyder model; in both
frameworks we find that the collapse is stabilized to an asymptotically static
state above the horizon, and the singularity is removed. In the Newtonian case,
by requiring the Newtonian approximation to be valid, we find lower bounds of
the order of unity (in Planck units) for the deformation parameter of the
modified algebra. We then study the behaviour of small perturbations on the
non-singular collapsing backgrounds, and find that for certain range of the
parameters (the polytropic index for the Newtonian case and the sound velocity
in the relativistic setting) the collapse is stable to perturbations of all
scales, and the non-singular super-Schwarzschild configurations have physical
meaning.
