Let K be a discretely valued field with ring of integers OK\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}_K$$\end{document} with perfect residue field. Let K(x) be the rational function field in one variable. Let POK1\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {P}}^1_{\mathcal {O}_K}$$\end{document} be the standard smooth model of PK1\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {P}}^1_K$$\end{document} with coordinate x. Let f(x)∈OK[x]\documentclass[12pt]{minimal}
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\begin{document}$$f(x) \in \mathcal {O}_K[x]$$\end{document} be a squarefree polynomial with corresponding divisor of zeroes div0(f)\documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{div}\,}}_0(f)$$\end{document} on POK1\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {P}}^1_{\mathcal {O}_K}$$\end{document}. We give an explicit description of the minimal embedded resolution Y\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Y}$$\end{document} of the pair (POK1,div0(f))\documentclass[12pt]{minimal}
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\begin{document}$$({\mathbb {P}}^1_{\mathcal {O}_K}, {{\,\mathrm{div}\,}}_0(f))$$\end{document} by using Mac Lane’s theory to write down the discrete valuations on K(x) corresponding to the irreducible components of the special fiber of Y\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {Y}$$\end{document}.
