Constructing Involutive Tableaux with Guillemin Normal Form
arXiv:1410.7593 · doi:10.3842/SIGMA.2015.053
Abstract
Involutivity is the algebraic property that guarantees solutions to an analytic and torsion-free exterior differential system or partial differential equation via the Cartan-Kähler theorem. Guillemin normal form establishes that the prolonged symbol of an involutive system admits a commutativity property on certain subspaces of the prolonged tableau. This article examines Guillemin normal form in detail, aiming at a more systematic approach to classifying involutive systems. The main result is an explicit quadratic condition for involutivity of the type suggested but not completed in Chapter IV, §5 of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and characterizes involutive tableaux.
This article co-evolved with "Degeneracy of the Characteristic Variety," arXiv:1410.6947 and most notation is shared. However, be aware that the meaning of the indices i,j,k,l and the space Y is not the same between these articles