Projecting dynamical systems via a support bound

For a polynomial dynamical system, we study the problem of computing the minimal differential equation satisfied by a chosen coordinate (in other words, projecting the system on the coordinate). This problem can be viewed as a special case of the general elimination problem for systems of differential equations and appears in applications to modeling and control. We give a bound for the Newton polytope of such minimal equation. Our bound depends on the dimension of the model and the degrees $d$ and $D$ of the polynomials defining the dynamics of the chosen coordinate and the remaining coordinates, respectively. We show that our bound is sharp if $d \leqslant D$ or the model is planar. We further use this bound to design an algorithm for computing the minimal equation following the evaluation-interpolation paradigm. We demonstrate that our implementation of the algorithm can tackle problems which are out of reach for the state-of-the-art software for differential elimination.