Simultaneous small fractional parts of polynomials

Abstract: 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 $c/k$.