  We introduce notions of vector field and its (discrete time) flow on a chain
complex. The resulting dynamical systems theory provides a set of tools with a
broad range of applicability that allow, among others, to replace in a
canonical way a chain complex with a "smaller" one of the same homotopy type.
As applications we construct in an explicit, canonical, and symmetry-preserving
fashion a minimal free resolution for every toric ring and every monomial
ideal. Our constructions work in all characteristics and over any base field. A
key subtle new point is that in certain finitely many positive characteristics
(which depend on the object that is being resolved) a transcendental extension
of the base field is produced before a resolution is obtained, while in all
other characteristics the base field is kept unchanged. In the monomial case we
show that such a transcendental base field extension cannot in general be
avoided, and we conjecture that the same holds in the toric case.
