The multi-bracket of m+1 differential operators on m unknown functions is an alternating sum of non-commutative determinants of their linearizations; for m=1 it collapses to the classical Jacobi-Mayer bracket, and to the commutator when the operators are linear. Vanishing of every multi-bracket of the operators defining a PDE system, taken modulo the differential ideal that system generates, is always necessary for formal integrability, and becomes sufficient for systems of generalized complete intersection type. Students learn to read compatibility as an explicit algebraic identity between operators rather than as the outcome of Cartan's prolongation-projection search, and to see why the Buchsbaum-Rim resolution of the symbolic module is what makes the obstructions computable.
Kruglikov and Lychagin define a multi-bracket of m+1 differential operators acting on m unknown functions: an alternating sum of non-commutative determinants of the operators' linearizations, extende…