Conductors and minimal discriminants of hyperelliptic curves: A comparison in the tame case

Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $mathrm{char} k > 2g+1$. Let $S = mathrm{Spec} R$. Let $X$ be the minimal proper regular model of $C$ over $S$. Let $mathrm{Art} (C/K)$ denote the Artin conductor of the $S$-scheme $X$ and let $ u (Delta_C)$ denote the minimal discriminant of $C$. We prove that $-mathrm{Art} (C/K) leq u (Delta_C)$. The key ingredients are a combinatorial refinement of the discriminant introduced in this paper (called the metric tree) and a recent refinement of Abhyankars inversion formula for studying plane curve singularities. We also prove combinatorial restrictions for $-mathrm{Art} (C/K) = u (Delta_C)$.