Geodesic
Constructing Involutive Tableaux with Guillemin Normal Form
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: involutivity; tableau; symbol; exterior differential systems. 
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Abraham D. Smith. Constructing Involutive Tableaux with Guillemin Normal Form. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a52/

[1] Bryant R. L., Chern S. S., Gardner R. B., Goldschmidt H. L., Griffiths P. A., Exterior differential systems, Mathematical Sciences Research Institute Publications, 18, Springer-Verlag, New York, 1991 http://library.msri.org/books/Book18/MSRI-v18-Bryant-Chern-et-al.pdf | DOI | MR | Zbl

[2] Carlson J., Green M., Griffiths P., “Variations of Hodge structure considered as an exterior differential system: old and new results”, SIGMA, 5 (2009), 087, 40 pp., arXiv: 0909.2201 | DOI | MR | Zbl

[3] Guillemin V., “Some algebraic results concerning the characteristics of overdetermined partial differential equations”, Amer. J. Math., 90 (1968), 270–284 | DOI | MR | Zbl

[4] Guillemin V. W., Quillen D., Sternberg S., “The integrability of characteristics”, Comm. Pure Appl. Math., 23 (1970), 39–77 | DOI | MR | Zbl

[5] Kruglikov B., Lychagin V., “Spencer $\delta$-cohomology, restrictions, characteristics and involutive symbolic PDEs”, Acta Appl. Math., 95 (2007), 31–50, arXiv: math.DG/0503124 | DOI | MR | Zbl

[6] Malgrange B., “Cartan involutiveness = Mumford regularity”, Commutative Algebra (Grenoble/{L}yon, 2001), Contemp. Math., 331, Amer. Math. Soc., Providence, RI, 2003, 193–205 | DOI | MR | Zbl

[7] Quillen D. G., Formal properties of over-determined systems of partial differential equations, Ph.D. Thesis, Harvard University, 1964 | Zbl

[8] Smith A. D., Degeneracy of the characteristic variety, arXiv: 1410.6947

[9] Yang D., Involutive hyperbolic differential systems, Mem. Amer. Math. Soc., 68, no. 370, 1987, xii+93 pp. | DOI | MR