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A relationship between Puiseux series satisfying an ordinary differentialequation corresponding to a polynomial dynamical system and degrees ofirreducible invariant algebraic curves is studied. A bound on the degrees ofirreducible invariant algebraic curves for a wide class of polynomial dynamicalsystems is obtained. It is demonstrated that the Puiseux series near infinitycan be used to find irreducible algebraic curves explicitly. As an example, allirreducible invariant algebraic curves for the famous FitzHugh-Nagumo systemare obtained.

From Puiseux series to invariant algebraic curves: the FitzHugh-Nagumo model