发表机构
Weizmann Institute of Science; Massachusetts Institute of Technology（魏茨曼科学研究所; 麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对多项式根式线性组合的实零点数目，将相关上界改进为多项式形式，回答了Alon的问题，还给出了高斯混合模型临界点数目的多项式上界及双曲平面特征函数的关联关系。
AI 中文摘要
我们针对形如 $f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{\alpha_k}$ 的函数的实零点数目获得上界，其中 $c_k, \alpha_k \in \mathbb{R}$，且每个 $P_k$ 是次数至多为 $d$ 的实多项式，在区间 $I\subset \mathbb{R}$ 上非负。我们将此前已知的区间 $I$ 上根的指数上界改进为关于 $n$ 为多项式、关于 $d$ 为线性且与指数 $\alpha_k$ 无关的上界。对于实数域上正二次多项式的平方根的线性组合，我们证明了线性上界 $2n$，回答了 N. Alon 的一个问题。对该论证的修改为 A. Gabrielov、D. Novikov 和 B. Shapiro 与 Maxwell 猜想相关的问题给出了线性上界。本文描述了两种独立方法：一般情形下的初等常微分方程方法，该方法还给出了一维高斯混合模型临界点数目的多项式上界；针对正二次多项式情形的偏微分方程方法，该方法将问题与穿孔双曲平面上满足 $\Delta u + \lambda u = 0$ 的解的节点域数目关联起来。作为第二种方法的副产品，我们描述了 $\mathbb{R}^3\setminus\{(x,0,0)\}$ 上轴对称调和函数与双曲平面上特征值为 $1/4$ 的 Laplace-Beltrami 特征函数之间的奇特关系。
英文摘要
We obtain upper bounds for the number of real zeros of functions of the form $$ f(x) = \sum_{k=1}^{n} c_k \bigl(P_k(x)\bigr)^{α_k}, $$ where $c_k, α_k \in \mathbb{R}$ and each $P_k$ is a real polynomial of degree at most $d$ that is non-negative on an interval $I\subset \mathbb{R}$. We improve previously known exponential upper bounds for the number of roots on $I$ to bounds that are polynomial in $n$, linear in $d$, and independent of the exponents $α_k$. For linear combinations of square roots of positive quadratic polynomials on $\mathbb{R}$ we prove the linear bound $2n$, answering a question of N.~Alon. A modification of the argument yields a linear bound for a question of A.~Gabrielov, D.~Novikov, and B.~Shapiro related to Maxwell's conjecture. The article describes two independent approaches: an elementary ODE method in the general case, which also gives a polynomial bound for the number of critical points of one dimensional Gaussian mixtures, and a PDE method for the case of positive quadratic polynomials, which connects the problem to the number of nodal domains of solutions to $Δu + λu = 0$ on the punctured hyperbolic plane. As a byproduct of the second approach, we describe a curious relation between axially symmetric harmonic functions on $\mathbb{R}^3\setminus\{(x,0,0)\}$ and Laplace-Beltrami eigenfunctions on the hyperbolic plane with eigenvalue $1/4$.
