Computation of Koszul homology and application to involutivity of partial differential systems

Abstract

The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OREMORPHISMS package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples.