Characterizing equivalent discrete Morse functions

We get a characterization theorem for equivalent discrete Morse functions defined on simplicial complexes in terms of their gradient vector field. As a consequence, we also characterize them in the 1-dimensional case by using critical elements. The authors are partially supported by the P.A.I. project (Junta de Andalucia, SPAIN, 2009/FQM-189 and 327) and by the MEC-FEDER grants MTM2007-61284 and MTM2007-65726 (MEC, SPAIN, 2007).