Let k be a d-local field such that the corresponding 1-local field \(k^{(d-1)}\) is a p-adic field and C a curve over k. Let K be the function field of C. We prove that for each \(n,m \in {\textbf{N}},\) and hypersurface Z of \({\textbf{P}}^n_K\) with degree m such that \(m^{d+1} \le n,\) the \((d+1)\) -th Milnor \({\textrm{K}}\) -theory group is generated by the images norms of finite extension L of K such that Z admits an L-point. Let \(j \in \{1,\ldots , d\}.\) When C admits a point in an extension l/k that is not i-ramified for every \(i \in \{1,\ldots , d-j\}\) we generalise this result to hypersurfaces Z of \({\textbf{P}}_K^n\) with degree m such that \(m^{j+1} \le n.\) In order to prove these results we give a description of the Tate–Shafarevich group in terms of the combinatorics of the special fibre of certain models of the curve C.
