关于多项式表示为指数函数线性组合的稀疏性
On sparsity of representations of polynomials as linear combinations of exponential functions
摘要：
给定一个整数$g$以及给定的一些整数$m$（充分大）和$c_1,\dots, c_m$，我们证明了所有非负整数$n\le M$的个数满足存在非负整数$k_1,\dots, k_m$使得$$n^2=\sum_{i=1}^m c_i g^{k_i}$$是$o\left(\left(\log M \right)^{m-1/2}\right)$的性质.我们也得到了一个类似的界限时，处理更一般的不等式$$\left|Q(n)-\sum_{i=1}^m c_iλ^{k_i}\right|\le B,$$，其中$Q\in {\mathbb C}[X]$，也$λ\in {\mathbb C}$（而$B$是一个实数）。
英文摘要：
Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots, c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots, k_m$ such that $$n^2=\sum_{i=1}^m c_i g^{k_i}$$ is $o\left(\left(\log M \right)^{m-1/2}\right)$. We also obtain a similar bound when dealing with more general inequalities $$\left|Q(n)-\sum_{i=1}^m c_iλ^{k_i}\right|\le B,$$ where $Q\in {\mathbb C}[X]$ and also $λ\in {\mathbb C}$ (while $B$ is a real number).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
