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An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviours are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero ($WR_0$) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding $WR_0$ realizations of polynomial dynamical systems, whenever such realizations exist.