The valuation of the discriminant of a hypersurface

Let $R$ be a discrete valuation ring, with valuation $v\text{\hspace{0.17em}⁣}:R\twoheadrightarrow {\mathbb{Z}}_{\ge 0}\cup \{\mathrm{\infty }\}$ and residue field $k$. Let $H$ be a hypersurface $\mathrm{Proj}(R[x_0,\dots ,x_n]\mathrm{/}⟨f⟩)$. Let $H_k$ be the special fiber, and let $(H_k{)}_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ be its singular subscheme. Let $\mathrm{\Delta }(f)$ be the discriminant of $f$. We use Zariski’s main theorem and degeneration arguments to prove that $v(\mathrm{\Delta }(f))=1$ if and only if $H$ is regular and $(H_k{)}_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}$ consists of a nondegenerate double point over $k$. We also give lower bounds on $v(\mathrm{\Delta }(f))$ when $H_k$ has multiple singularities or a positive-dimensional singularity.