  We present a systematic methodology to determine and locate analytically
isolated periodic points of discrete and continuous dynamical systems with
algebraic nature. We apply this method to a wide range of examples, including a
one-parameter family of counterexamples to the discrete Markus-Yamabe
conjecture (La Salle conjecture); the study of the low periods of a
Lotka-Volterra-type map; the existence of three limit cycles for a piece-wise
linear planar vector field; a new counterexample of Kouchnirenko's conjecture;
and an alternative proof of the existence of a class of symmetric central
configuration of the $(1+4)$-body problem.
