Simultaneous small fractional parts of polynomials

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 $c/k$.