Loewy Decomposition of Linear Differential Equations

Outline

1. Loewys Results for Linear ODE's.
2. Basic Concepts of Differential Algebra.
3. Equations with Finite-Dimensional Solution Space.
4. Decomposing Second- and Third-Order Equations.
5. Summary and Outlook.
6. Loewy Decomposition with ALLTYPES.

References

1. A. Loewy, "Ueber vollstaendig reduzible lineare homogene Differentialgleichungen", Mathematische Annalen, 56, page 89-117 (1906).
2. Z. Li, F. Schwarz, S. Tsarev, "Factoring Zero-dimensional Ideals of Linear Partial Differential Operators", Proceedings of the ISSAC'02, ACM Press. T. Mora, Ed., 168-175, 2002.
3. D. Grigoriev, F. Schwarz, "Factoring and Solving Linear Partial Differential Equations", Computing 73, page 179-197 (2004).

Gröbner Bases: A Sampler of Recent Development

This list of topics is tentative—the tutorial may cover slightly different topics.

Outline

1. Review of Basic Material about Gröbner Bases
2. The Geometry of Elimination via Gröbner Bases
3. Some Recent Developments in the Theory and Applications of Gröbner Bases

References

1. B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aeq. Math. 3 (1970), 374–383. (English Translation pp. 535–545).
2. B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklasse-ringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, W. Gröbner, advisor, Innsbruck, 1965.
3. B. Buchberger, Gröbner bases: an algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory – Progress, Directions and Open Problems in Multidimensional Systems (N. K. Bose, editior), Reidel Publishing Company, Dordrecht, 1985, 184–232. (Second edition: Multidimensional Systems Theory and Applications (N. K. Bose, editor) Kluwer, Dordrecht, 2003, 89–128.)
4. B. Buchberger and F. Winkler (editors), Gröbner Bases and Applications, Cambridge U. Press, Cambridge, 1998.
5. A. Craw, D. Maclagan and R. Thomas, Moduli of McKay quiver representations II: Gröbner Basis techniques, arXiv:math.Ag/0611840.
6. D. Cox, J. Little and D. O’Shea, Ideals, Varieties and Algorithms, Third edition, Springer, New York, 2007.
7. D. Cox, J. Little and D. O’Shea, Using Algebraic Geometry, Second edition, Springer, New York, 2005.
8. J.-C. Faugère, A new efficient algorithm for computing Gröbner bases (F4), J. Pure Appl. Algebra 139 (1999), 61–88.
9. J.-C. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zeros (F5), in Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2002, 75–83.
10. J.-C. Faugère, M. Hering and J. Phan, The membrane inclusions curvature equations, Adv. in Appl. Math. 31 (2003), 643–658.
11. C. Hillar and T. Windfeldt, Algebraic characterization of uniquely vertex colorable graphs, arXiv:math.CO/0606565
12. K. Fukada, N. Jensen, N. Lauritzen and R. Thomas, The generic Gröbner walk, J. Symbolic Comput. 42 (2007), 298–312.
13. M. Manubens and A. Montes, Minimal canonical comprehensive Gröbner system, arXiv:math.AG/0611948.
14. A. Montes and T. Recio, Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems, arXiv:math.AG/0703483.
15. P. Schauenberg, A Gröbner-based treatment of elimination theory for affine varieties, J. Symbolic Comput., to appear.
16. RICAM, Special Semester on Gröbner Bases 2006. Information available at the web site http://www.ricam.oeaw.ac.at/specsem/srs/groeb

Some Recent Progress in Symbolic Linear Algebra and Related Questions

More information available at http://perso.ens-lyon.fr/gilles.villard/linalg.html.