Summary: We consider the Hypergraph-\(k\)-Cut problem. The input consists of a hypergraph \(G = ( V , E )\) with nonnegative hyperedge-costs \(c : E \to \mathbb{R}_+\) and a positive integer \(k\). The objective is to find a minimum cost subset \(F \subseteq E\) such that the number of connected components in \(G - F\) is at least \(k\). An alternative formulation of the objective is to find a partition of \(V\) into \(k\) nonempty sets \(V_1 , V_2 , \ldots , V_k\) so as to minimize the cost of the hyperedges that cross the partition. Graph-\(k\)-Cut, the special case of Hypergraph-\(k\)-Cut obtained by restricting to graph inputs, has received considerable attention. Several different approaches lead to a polynomial-time algorithm for Graph-\(k\)-Cut, when \(k\) is fixed, starting with the work of Goldschmidt and Hochbaum (Math of OR, 1994). In contrast, it is only recently that a randomized polynomial time algorithm for Hypergraph-\(k\)-Cut was developed (Chandrasekaran, Xu, Yu, Math Programming, 2019) via a subtle generalization of Karger’s random contraction approach for graphs. In this work, we develop the first deterministic algorithm for Hypergraph-\(k\)-Cut that runs in polynomial time for any fixed \(k\). We describe two algorithms both of which are based on a divide and conquer approach. The first algorithm is simpler and runs in \(n^{O ( k^2 )} m\) time while the second one runs in \(n^{O ( k )} m\) time, where \(n\) is the number of vertices and \(m\) is the number of hyperedges in the input hypergraph. Our proof relies on new structural results that allow for efficient recovery of the parts of an optimum \(k\)-partition by solving minimum \((S,T)\)-terminal cuts. Our techniques give new insights even for Graph-\(k\)-Cut.