New bounds for the support of input-output equations in differential-algebraic systems

2026-09-03

AI Summary

本文研究了确定由微分-代数系统导出的输入输出方程的最小多项式问题，通过设定多项式的度上界和定义包含其牛顿多面体的不等式组来解决。

Abstract

Given a polynomial dynamical system $\mathbf{x}'=\mathbf{f}(\mathbf{x},\mathbf{u})$ together with an observation function $y=g(\mathbf{x},\mathbf{u})$, where $\mathbf{x}=(x_1,\ldots,x_n)$, $\mathbf{u}=(u_1,\ldots,u_m)$ and $y$ are differential variables, and $\mathbf{f}=(f_1,\ldots,f_n)$, $g$ are polynomials with coefficients in a differential field, we study the problem of determining a minimal polynomial differential equation satisfied by the inputs $\mathbf{u}$ and the output $y$ which follows as a differential consequence of the system. We provide a characterization of a finite superset of the set of monomials appearing with non-zero coefficients in this input-output equation. Specifically, we establish an upper bound for the degree of the minimal polynomial and a family of inequalities that define a polytope containing its Newton polytope. These results extend recent work by Mukhina and Pogudin for systems with constant parameters, and enable the use of evaluation-interpolation techniques for the efficient computation of such eliminant polynomials.

Problem

• differential-algebraic systems
• input-output equations
• minimal polynomial
• Newton polytope

Innovation

• differential-algebraic systems
• input-output equations
• Newton polytope
• evaluation-interpolation techniques
• polynomial differential equation