Explicit minimal embedded resolutions of divisors on models of the projective line

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