Simultaneously Small Fractional Parts of Polynomials

Let f₁,,fₖ ∈ R[X] be polynomials of degree at most d with f₁(0)==fₖ(0)=0. We show that there is an $n<x$ such that \|fᵢ(n)\| x-1/10.5kd(d-1)+o(1) for all 1≤ i≤ k. This improves on an earlier result of Maynard, who obtained the same exponent dependency on k but not on d.