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Projecting dynamical systems via a support bound

Yulia Mukhina, Gleb Pogudin

TL;DR

This work addresses projecting polynomial dynamical systems onto a single coordinate by computing the minimal differential equation satisfied by that coordinate. It reduces differential elimination to polynomial elimination and derives a tight bound for the Newton polytope of the minimal equation, depending only on $d = \deg g_1$ and $D = \max_{i\ge2} \deg g_i$, with sharpness in many cases (e.g., $d \le D$ or $n=2$). Leveraging this bound, the authors design an evaluation–interpolation algorithm that recovers the minimal equation from sampled points, enabling elimination tasks beyond the reach of existing software; a Julia implementation demonstrates practical scalability. They also discuss the limits of the bound, experimental accuracy, and potential refinements via tropical implicitization and mixed fiber polytopes. Overall, the approach provides a concrete, scalable path to differential elimination in polynomial dynamical systems with broad modeling and control applications.

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.

Projecting dynamical systems via a support bound

TL;DR

This work addresses projecting polynomial dynamical systems onto a single coordinate by computing the minimal differential equation satisfied by that coordinate. It reduces differential elimination to polynomial elimination and derives a tight bound for the Newton polytope of the minimal equation, depending only on and , with sharpness in many cases (e.g., or ). Leveraging this bound, the authors design an evaluation–interpolation algorithm that recovers the minimal equation from sampled points, enabling elimination tasks beyond the reach of existing software; a Julia implementation demonstrates practical scalability. They also discuss the limits of the bound, experimental accuracy, and potential refinements via tropical implicitization and mixed fiber polytopes. Overall, the approach provides a concrete, scalable path to differential elimination in polynomial dynamical systems with broad modeling and control applications.

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.
Paper Structure (20 sections, 31 theorems, 128 equations, 1 figure, 5 tables, 2 algorithms)

This paper contains 20 sections, 31 theorems, 128 equations, 1 figure, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Bound for the support and its sharpness
  5. Discussion of the bound
  6. Experimental evaluation of the bound's accuracy
  7. Potential and limitations of the present approach
  8. On the alternative approach via tropical implicitization
  9. Proofs: general facts and notation
  10. Proofs: the bound for the support
  11. Weighted Bézout bound
  12. Bound for the support in a general form
  13. Proof of Theorem \ref{['theorem_general_specialized']}
  14. Proofs: sharpness of the bound
  15. From sharpness to generic sharpness
  16. ...and 5 more sections

Key Result

Proposition 1

The prime ideal id is uniquely determined by its minimal polynomial $f_{\min}$. More precisely:

Figures (1)

  • Figure 1: Newton polytopes predicted by Theorem \ref{['theorem_general_specialized']} for the planar case $n = 2$

Theorems & Definitions (70)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1: pogudin2023differential
  • Example 1
  • Example 2
  • Theorem 1: Bound for the support
  • Theorem 2: Generic sharpness in the planar case
  • ...and 60 more