Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

We prove a conductor-discriminant inequality for all hyperelliptic curves defined over discretely valued fields $K$ with perfect residue field of characteristic not $2$. Specifically, if such a curve is given by $y^2 = f(x)$ with $f(x) \in \mathcal{O}_K[x]$, and if $\mathcal{X}$ is its minimal regular model over $\mathcal{O}_K$, then the negative of the Artin conductor of $\mathcal{X}$ (and thus also the number of irreducible components of the special fiber of $\mathcal{X}$) is bounded above by the valuation of $\mathrm{disc}\ (f)$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $f$. This generalizes earlier work of Ogg, Saito, Liu, and the second author. The proof relies on using so-called Mac Lane valuations to resolve singularities of arithmetic surfaces, a technique that was recently introduced by Wewers and the first author.