微分代数系统中输入输出方程支集的新边界
New bounds for the support of input-output equations in differential-algebraic systems
- Universidad de Buenos Aires（布宜诺斯艾利斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对微分代数系统，建立了输入输出方程最小多项式的次数上界与相关多面体不等式，推广了前人工作，为高效计算消元多项式提供了方法支撑。
AI中文摘要：
给定多项式动力系统x'=f(x,u)以及观测函数y=g(x,u)，其中x=(x₁,…,xₙ)、u=(u₁,…,uₘ)和y为微分变量，f=(f₁,…,fₙ)、g为系数取自微分域的多项式，我们研究确定输入u与输出y所满足的最小多项式微分方程的问题，该方程是系统的微分推论。我们对该输入输出方程中系数非零的单项式集合的有限超集给出了刻画，具体而言，我们建立了最小多项式次数的上界，以及定义包含其牛顿多面体的多面体的一组不等式。这些结果推广了Mukhina和Pogudin针对含常数参数系统的近期工作，使评估插值技术可用于高效计算此类消元多项式。
英文摘要：
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.
