标题： 稳定系统中的振幅最大化，Schur 正性，以及关于多项式插值的一些猜想 显示英文标题

标题： Amplitude maximization in stable systems, Schur positivity, and some conjectures on polynomial interpolation

摘要： 对于$r > 0$和整数$t \ge n > 0$，我们考虑以下问题：在所有具有特征根在圆盘$\{z \in \mathbb{C}: |z| \le r\}$内的任意齐次线性差分方程的复数解$x = (x_0, x_1, \dots)$中，最大化时间$t$的振幅$|x_t|$，这些方程的阶数为$n$，初始值$x_0, \dots, x_{n-1}$在单位圆内。 我们发现，对于任何三元组$t,n,r$，最大值在边界圆上的重根处达到；特别是，这表明峰值振幅$\sup_{t \ge n} |x_t|$可以通过研究具有特征多项式$(z-r)^n$的唯一方程来显式最大化。 此外，共相根配置的最优性适用于以原点为中心的多圆盘。 为了证明这个结果，我们首先将问题简化为关于单项式的某种插值问题，然后通过利用对称函数理论并识别相关的 Schur 正性结构来解决后者。 我们还讨论了对更一般的 Reinhardt 域的影响。 最后，我们研究了从单位圆内 $n/2$ 对共轭复数点的值估计实整函数导数的问题。 我们提出了关于单项式$z^n$极端性的猜想，并用 Schur 多项式重新表述它们。 显示英文摘要

摘要： For $r > 0$ and integers $t \ge n > 0$, we consider the following problem: maximize the amplitude $|x_t|$ at time $t$, over all complex solutions $x = (x_0, x_1, \dots)$ of arbitrary homogeneous linear difference equations of order $n$ with the characteristic roots in the disc $\{z \in \mathbb{C}: |z| \le r\}$, and with initial values $x_0, \dots, x_{n-1}$ in the unit disc. We find that for any triple $t,n,r$, the maximum is attained with coinciding roots on the boundary circle; in particular, this implies that the peak amplitude $\sup_{t \ge n} |x_t|$ can be maximized explicitly, by studying a unique equation with the characteristic polynomial $(z-r)^n$. Moreover, the optimality of the cophase root configuration holds for origin-centered polydiscs. To prove this result, we first reduce the problem to a certain interpolation problem over monomials, then solve the latter by leveraging the theory of symmetric functions and identifying the associated Schur positivity structure. We also discuss the implications for more general Reinhardt domains. Finally, we study the problem of estimating the derivatives of a real entire function from its values at $n/2$ pairs of complex conjugate points in the unit disc. We propose conjectures on the extremality of the monomial $z^n$, and restate them in terms of Schur polynomials.