多项式的同时小分数部分
Simultaneous small fractional parts of polynomials
摘要：
设$f_1,\dots,f_k\in\mathbb{R}[X]$是次数至多为$d$且具有$f_1(0)=\dots=f_k(0)=0$的多项式。证明了存在一个整数$n<x$，使得对于所有的$1\le i\le k$和某些常数$c=c(d)$，分数部分$\|f_i(n)\|\ll x^{c/k}$只依赖于$d$ .这在$k$方面基本上是最优的，并且改进了Schmidt的早期结果，Schmidt用$c/k^2$代替$c/k$显示了相同的结果。
英文摘要：
Let $f_1,\dots,f_k\in\mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an integer $n<x$ such that the fractional parts $\|f_i(n)\|\ll x^{c/k}$ for all $1\le i\le k$ and for some constant $c=c(d)$ depending only on $d$. This is essentially optimal in the $k$-aspect, and improves on earlier results of Schmidt who showed the same result with $c/k^2$ in place of $c/k$.
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