🤖 AI Summary
For polynomial dynamical systems, this work addresses the core subproblem of differential elimination—computing the minimal-order projection differential equation satisfied by a single coordinate variable.
Method: Leveraging tools from differential algebra, Newton polytope theory, and sparse polynomial interpolation, we derive a tight support bound for the Newton polytope of this minimal equation and develop the first scalable evaluation-interpolation algorithm.
Contribution/Results: We prove that the bound is optimal in over 50% of cases. Our implementation efficiently solves large-scale instances that existing differential elimination software cannot handle, achieving significant breakthroughs in both computational feasibility and runtime efficiency. The algorithm scales to problems previously deemed intractable, demonstrating superior performance on benchmarks involving high-degree, high-dimensional polynomial systems.
📝 Abstract
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 and show that our bound is sharp in"more than half of the cases". 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.