Simultaneously Small Fractional Parts of Polynomials

Let $f_1,\dots ,f_k\in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\cdots =f_k(0)=0$. We show that there is an $n<x$ such that $\mathrm{\parallel }{f}_i(n)\mathrm{\parallel }\ll x^{-1\mathrm{/}10.5kd(d-1)+o(1)}$ 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 $d$.