Constructing Involutive Tableaux with Guillemin Normal Form
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-K\"ahler 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, \S 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and characterizes involutive tableaux.
Keywords
Cite
@article{arxiv.1410.7593,
title = {Constructing Involutive Tableaux with Guillemin Normal Form},
author = {Abraham D. Smith},
journal= {arXiv preprint arXiv:1410.7593},
year = {2015}
}
Comments
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