Jacobi identities for Wronskian determinants over multidimension

The generalised Wronskian of differential order $k⩾1$ for $N$ functions $f_1$, $\dots$, $f_N$ in $d⩾1$ independent variables $x^1$, $\dots$, $x^d$ is the determinant of the matrix with these functions’ derivatives ${\mathrm{\partial }}^{\mathrm{\mid }{\sigma }_i\mathrm{\mid }}{f}_j\mathrm{/}\mathrm{\partial }(x^1{)}^{{\sigma }_i^1}\cdots \mathrm{\partial }(x^d{)}^{{\sigma }_i^d}$ (of orders $0⩽\mathrm{\mid }{\sigma }_i\mathrm{\mid }⩽k$), where the multi-indices ${\sigma }_i$ mark (all or part of) fibre variables $u_{{\sigma }_i}$ in the $k$th jet space $J^k({\mathbb{R}}^d\to \mathbb{R})$. We prove that these (in)complete Wronskians – provided that their lowest-order parts are complete at differential orders $\mathrm{\ell }⩽1$ – over the $d$-dimensional base satisfy the table of bi-linear, Jacobi-type identities for Schlessinger–Stasheff’s strongly homotopy Lie algebras.