  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 $n<x$ such that
$\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o(1)}$ for all $1\le i\le k$. This improves on
an earlier result of Maynard, who obtained the same exponent dependency on $k$
but not on $d$.
