Jacobi identities for Wronskian determinants over multidimension
Arthemy V. Kiselev
TL;DR
The paper addresses extending Wronskian determinants to functions of $d$ variables and establishing their Jacobi-type identities within strongly homotopy Lie algebras. It defines complete and incomplete multidimensional Wronskians $W^{d\geq1}_{k\geq1}$ with $N=\binom{d+k}{d}$ and proves that the Jacobi relations $W^{d\geq1}_{k,out}[W^{d\geq1}_{\ell,in}]=0$ hold under a completeness condition for first-order derivatives, generalizing known 1D results to higher dimensions. It also analyzes incomplete cases, provides precise admissibility criteria, and presents counterexamples to show necessity of the completeness assumption, while outlining open problems about the integral structure of the deformed algebra generated by these Wronskians. Overall, the work connects multidimensional Wronskians, jet-space derivatives, and strongly homotopy Lie algebra deformations, contributing a structured algebraic framework for higher-dimensional linear independence phenomena and symmetry analysis.
Abstract
The generalised Wronskian of differential order $k\geqslant 1$ for $N$ functions $f_1$, $\ldots$, $f_N$ in $d\geqslant 1$ independent variables $x^1$, $\ldots$, $x^d$ is the determinant of the matrix with these functions' derivatives $\partial^{|σ_i|} f_j / \partial (x^1)^{σ_i^1}\cdots \partial (x^d)^{σ_i^d}$ (of orders $0 \leqslant |σ_i| \leqslant k$), where the multi-indices $σ_i$ mark (all or part of) fibre variables $u_{σ_i}$ in the $k$th jet space $J^k\bigl(\mathbb{R}^d\to\mathbb{R}\bigr)$. We prove that these (in)complete Wronskians -- provided that their lowest-order parts are complete at differential orders $\ell\leqslant 1$ -- over the $d$-dimensional base satisfy the table of bi-linear, Jacobi-type identities for Schlessinger--Stasheff's strongly homotopy Lie algebras.