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Milnor $K$-groups and function fields of hypersurfaces in positive characteristic

Let $X$ be an integral affine or projective hypersurface over a field $F$ of characteristic $p>0$, and let $F(X)$ denote its function field. In a recent article, Dolphin and Hoffmann obtained an explicit description of the kernel of the natural restriction homomorphism between the rings of absolute Kähler differentials of $F$ and $F(X)$, respectively. In this note, we examine the possibility of deriving an analogous result for mod-$p$ Milnor $K$-theory using the Bloch-Gabber-Kato theorem.