Before revision:  The complexity of our algorithms is 2^{(\log(1/ \varepsilon \eps )/ \varepsilon \eps )^{O(d^2)}} n \log^4 n + 2^{O(d)} n \log^9 n, making a significant improvement over the state-of-the-art algorithm which runs in time n^{(d/ \varepsilon \eps )^{O(d)}}  .
After revision:  The complexity of our algorithms is 2^{(\log(1/ \varepsilon \eps )/ \eps )^{O(d^2)}} n \log^4 n + 2^{O(d)} n \log^9 n, making a significant improvement over the state-of-the-art algorithm which runs in time n^{(d/ \varepsilon \eps )^{O(d)}} .