Geodesic
Note on two compatibility criteria: Jacobi--Mayer bracket vs.\ differential Gr\"obner basis
Lobachevskii journal of mathematics, Tome 23 (2006), pp. 57-70.

Voir la notice de l'article provenant de la source Math-Net.Ru

We compare two compatibility criteria for overdetermined PDEs: one via geometric theory of differential equations and another via differential algebra approach. Whenever both are applicable, we show that the former is more effective, though in some very special cases they are equivalent.
Keywords: compatibility of differential equations, Jacobi bracket, Mayer bracket, differential Gröbner basis, Kolchin–Ritt algorithm. 
@article{LJM_2006_23_a2,
     author = {B. S. Kruglikov},
     title = {Note on two compatibility criteria: {Jacobi--Mayer} bracket vs.\ differential {Gr\"obner} basis},
     journal = {Lobachevskii journal of mathematics},
     pages = {57--70},
     publisher = {mathdoc},
     volume = {23},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2006_23_a2/}
}
TY  - JOUR
AU  - B. S. Kruglikov
TI  - Note on two compatibility criteria: Jacobi--Mayer bracket vs.\ differential Gr\"obner basis
JO  - Lobachevskii journal of mathematics
PY  - 2006
SP  - 57
EP  - 70
VL  - 23
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/LJM_2006_23_a2/
LA  - en
ID  - LJM_2006_23_a2
ER  - 
%0 Journal Article
%A B. S. Kruglikov
%T Note on two compatibility criteria: Jacobi--Mayer bracket vs.\ differential Gr\"obner basis
%J Lobachevskii journal of mathematics
%D 2006
%P 57-70
%V 23
%I mathdoc
%U http://geodesic.mathdoc.fr/item/LJM_2006_23_a2/
%G en
%F LJM_2006_23_a2
B. S. Kruglikov. Note on two compatibility criteria: Jacobi--Mayer bracket vs.\ differential Gr\"obner basis. Lobachevskii journal of mathematics, Tome 23 (2006), pp. 57-70. http://geodesic.mathdoc.fr/item/LJM_2006_23_a2/

[BW] T. Becker, V. Weispfenning, Gröbner bases. A computational approach to commutative algebra, Graduate Texts in Mathematics, 141, Springer-Verlag, 1993 | MR | Zbl

[BC] G. Bluman, J. D. Cole, “The general similarity solution of the heat equation”, J. Math. Mech., 18 (1968/69), 1025–1042 | MR

[CK] P. A. Clarkson, M. D. Kruskal, “New similarity reductions of the Boussinesq equation”, J. Math. Phys., 30:10 (1989), 2201–2213 | DOI | MR | Zbl

[F] G. Carrà-Ferro, “Gröbner bases and differential algebra”, Applied algebra, algebraic algorithms and error-correcting codes (Menorca, 1987), Lecture Notes in Comput. Sci., 356, Springer, Berlin, 1989, 129–140 | MR

[Gu] N. M. Gunter, Integration of PDEs of the First order, ONTI (Russian), Leningrad–Moscow, 1934

[H] E. Hubert, “Notes on triangular sets and triangulation-decomposition algorithms: I. Polynomial systems”, Symbolic and numerical scientific computation (Hagenberg, 2001), Lecture Notes in Comput. Sci., 2630, Springer, Berlin, 2003, 1–39 ; “II. Differential systems”, 40–87 | MR | Zbl | Zbl

[J] M. Janet, Leçons sur les systèms d'équations, Gauthier-Villers, Paris, 1929 | Zbl

[K] E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, 54, Academic Press, New York–London, 1973 | MR | Zbl

[KLV] I. S. Krasilschik, V. V. Lychagin, A. M. Vinogradov, Geometry of jet spaces and differential equations, Gordon and Breach, 1986

[KL1] B. S. Kruglikov, V. V. Lychagin, “Mayer brackets and solvability of PDEs – I”, Diff. Geom. and its Appl., 17 (2002), 251–272 | DOI | MR | Zbl

[KL2] B. S. Kruglikov, V. V. Lychagin, “Mayer brackets and solvability of PDEs – II”, Trans. Amer. Math. Soc., 358:3 (2006), 1077–1103 | DOI | MR | Zbl

[KL3] B. S. Kruglikov, V. V. Lychagin, “A compatibility criterion for systems of PDEs and generalized Lagrange–Charpit method”, Global Analysis and Applied Mathematics, International Workshop on Global Analysis, American Inst. of Physics Conference Proc., 729, no. 1, Amer. Inst. Phys., Melville, NY, 2004, 39–53 | MR | Zbl

[KL4] B. S. Kruglikov, V. V. Lychagin, “Multi-brackets of differential operators and compatibility of PDE systems”, C. R. Math. Acad. Sci. Paris, 342:8 (2006), 557–561 | MR | Zbl

[M1] E. L. Mansfield, Differential Gröbner bases, Ph.D. Thesis, University of Sidney, 1987

[M2] E. L. Mansfield, “A simple criterion for involutivity”, J. London Math. Soc. (2), 54:2 (1996), 323–345 | MR | Zbl

[RWB] G. J. Reid, A. D. Wittkopf, A. Boulton, “Reduction of systems of nonlinear partial differential equations to simplified involutive forms”, European J. Appl. Math., 7:6 (1996), 635–666 | DOI | MR | Zbl

[R] J. F. Ritt, Differential Algebra, A.M.S. Colloquium Publications, 33, New York, 1950 ; Dover Publ., 1966 | MR | Zbl

[S] D. C. Spencer, “Overdetermined systems of linear partial differential equations”, Bull. Amer. Math. Soc., 75 (1969), 179–239 | DOI | MR | Zbl

[V] W. V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, 2, Springer-Verlag, Berlin, 1998 | MR | Zbl