We solve the dual multijoint problem and prove the existence of so-called factorisations for arbitrary fields and multijoints of \(k_j\) -planes. More generally, we deduce a discrete analogue of a theorem due in essence to Bourgain and Guth. Our result is a universal statement which describes a property of the discrete wedge product without any explicit reference to multijoints and is stated as follows: Suppose that \(k_1 + \ldots + k_d = n\) . There is a constant \(C=C(n)\) so that for any field \(\mathbb {F}\) and for any finitely supported function \(S : \mathbb {F}^n \rightarrow \mathbb {R}_{\ge 0}\) , there are factorising functions \(s_{k_j} : \mathbb {F}^n\times {{\,\mathrm{{Gr}}\,}}(k_j, \mathbb {F}^n)\rightarrow \mathbb {R}_{\ge 0}\) such that \(\begin{aligned} \left( V_1 \wedge \cdots \wedge V_d\right) S\left( p\right) ^d \le \prod _{j=1}^d s_{k_j}\left( p, V_j\right) , \end{aligned}\) for every \(p\in \mathbb {F}^n\) and every tuple of planes \(V_j\in {{\,\mathrm{{Gr}}\,}}(k_j, \mathbb {F}^n)\) , and \(\begin{aligned} \sum _{p\in \pi _j} s(p, e(\pi _j)) =C \left| \left| S\right| \right| _d, \end{aligned}\) for every \(k_j\) -plane \(\pi _j\subset \mathbb {F}^n\) , where \(e(\pi _j)\in {{\,\mathrm{{Gr}}\,}}(k_j,\mathbb {F}^n)\) , is the translate of \(\pi _j\) that contains the origin and \(\wedge \) denotes the discrete wedge product.
