Abstract
An algorithm for factoring differential systems in characteristic p has been given by Cluzeau in [7]. It is based on both the reduction of a matrix called p-curvature and eigenring techniques. In this paper, we generalize this algorithm to factor partial differential systems in characteristic p. We show that this factorization problem reduces effectively to the problem of simultaneous reduction of commuting matrices. In the appendix, van der Put shows how to extend his classification of differential modules, used in the work of Cluzeau, to partial differential systems in positive characteristic.
with an appendix by M. van der Put: Classification of Partial Differential Modules in Positive Characteristic
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References
S. S. Abhyankar. Algebraic geometry for scientists and engineers. Mathematical Surveys and Monographs, vol. 35. AMS, 1990.
M. A. Barkatou. On the reduction of matrix pseudo-linear equations. Technical Report RR 1040, Rapport de Recherche de l’Institut IMAG, 2001.
M. A. Barkatou and E. Pflügel. An algorithm computing the regular singular formal solutions of a linear differential system. J. Symb. Comput. 28(4-5), 1999.
Y. A. Blinkov, C. F. Cid, V. T. Gerdt, W. Plesken and D. Robertz. The Maple package “Janet” II: Polynomial systems. In: Proceedings of the Workshop on Computer Algebra and Scientific Computing (CASC), Passau, 2003. http://wwwmayr.informatik.tu-muenchen.de/CASC2003/
L. Chambadal and J. L. Ovaert. Algèbre linéaire et algèbre tensorielle. Dunot Université, Paris, 1968.
A. Chambert-Loir. Théorèmes d’algébricité en géométrie diophantienne. Séminaire Bourbaki, exposé No. 886, Mars 2001.
T. Cluzeau. Factorization of differential systems in characteristic p. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 58–65. ACM Press, New York, 2003.
T. Cluzeau. Algorithmique modulaire des équations différentielles linéaires. Thèse de lÚniversité de Limoges, Septembre 2004.
T. Cluzeau and M. van Hoeij. A modular algorithm for computing the exponential solutions of a linear differential operator. J. Symb. Comput. 38(3): 1043–1076, 2004.
R. Corless, P. Gianni and B. Trager. A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 133–140. ACM Press, New York, 1997.
M. Giesbrecht and Y. Zhang. Factoring and decomposing Ore polynomials over \(\mathbb{F}_p (t)\) (t). In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp. 127–134. ACM Press, New York, 2003.
M. Hausdorf and M. Seiler. Involutives basis in MuPAD — Part I: Involutive divisions. MathPad 11(1): 51–56, 2002.
M. van Hoeij, J.-F. Ragot, F. Ulmer and J.-A. Weil. Liouvillian solutions of linear differential equations of order three and higher. J. Symb. Comput. 11: 1–17, 1998.
N. Jacobson. Lectures in abstract algebra II: Linear algebra. Graduate Texts in Mathematics, vol. 31. Springer, New York, 1953.
N. Jacobson. Basic algebra II. W. H. Freeman and Compagny, San Francisco, 1980.
M. Janet. Sur les systèmes aux dérivées partielles. Journal de Math., 8-ème série, III, 65–151, 1924.
M. Janet. Leçons sur les systèmes d’équations aux dérivées partielles. Cahiers Scientifiques, IV, Gauthiers-Villars, 1929.
N. Katz. Nilpotent connections and the monodromy theorem: Applications of a result of Turritin. Publ. Math. I. H. E. S. 39: 355–412, 1970.
N. Katz. A conjecture in the arithmetic theory of differential equations. Bull. Soc. Math. France 110: 203–239, 1982.
A. H. M. Levelt. The semi-simple part of a matrix. Algorithmen in de Algebra, 85–88, 1994. http://www.math.ru.nl/medewerkers/ahml/other.htm
Z. Li, F. Schwarz and S. P. Tsarev. Factoring zero-dimensional ideals of linear partial differential operators. In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pp. 168–175. ACM Press, New York, 2002.
Z. Li, F. Schwarz and S. P. Tsarev. Factoring systems of linear PDEs with finitedimensional solution spaces. J. Symb. Comput. 36(3-4): 443–471, 2003.
M. van der Put. Differential equations in characteristic p. Compositio Math. 97: 227–251, 1995.
M. van der Put. Modular methods for factoring differential operators. Manuscript, 1997.
M. van der Put and M. F. Singer. Galois theory of difference equations. Lecture Notes in Mathematics, vol. 1666. Springer, Berlin, 1997.
M. van der Put and M. F. Singer. Galois theory of linear differential equations. Grundlehren der mathematischen Wissenschaften, vol. 328. Springer, Berlin, 2003.
M. F. Singer. Testing reducibility of linear di-erential operators: A group theoretic perspective. Appl. Alg. Eng. Comm. Comp. 7(2): 77–104, 1996.
M. Wu. On the factorization of differential modules. In this volume.
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Barkatou, M.A., Cluzeau, T., Weil, JA. (2005). Factoring Partial Differential Systems in Positive Characteristic. In: Wang, D., Zheng, Z. (eds) Differential Equations with Symbolic Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7429-2_13
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DOI: https://doi.org/10.1007/3-7643-7429-2_13
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