  We provide new deterministic algorithms for the edge coloring problem, which
is one of the classic and highly studied distributed local symmetry breaking
problems. As our main result, we show that a $(2\Delta-1)$-edge coloring can be
computed in time $\mathrm{poly}\log\Delta + O(\log^* n)$ in the LOCAL model.
This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an
algorithm with a quasi-polylogarithmic dependency on $\Delta$. We further show
that in the CONGEST model, an $(8+\varepsilon)\Delta$-edge coloring can be
computed in $\mathrm{poly}\log\Delta + O(\log^* n)$ rounds. The best previous
$O(\Delta)$-edge coloring algorithm that can be implemented in the CONGEST
model is by Barenboim and Elkin [PODC '11] and it computes a
$2^{O(1/\varepsilon)}\Delta$-edge coloring in time $O(\Delta^\varepsilon +
\log^* n)$ for any $\varepsilon\in(0,1]$.
