×

zbMATH — the first resource for mathematics

Regular \(b\)-functions of \(D\)-modules. (English) Zbl 1160.32013
Summary: Let \(M\) be an algebraic \(D\)-module defined on an affine space \(X\) and \(Y\) be a linear submanifold of \(X\). We give an algorithm to determine if \(M\) is regular specializable along \(Y\), and to find, if so, its regular \(b\)-function. (\(M\) has a regular \(b\)-function by definition if and only if \(M\) is regular specializable.) We also prove that the \(A\)-hypergeometric system of Gelfand-Kapranov-Zelevinsky is always regular specializable along the origin.

MSC:
32C38 Sheaves of differential operators and their modules, \(D\)-modules
33C70 Other hypergeometric functions and integrals in several variables
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
35N10 Overdetermined systems of PDEs with variable coefficients
Software:
SINGULAR; Kan
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Laurent, Y.;  , Y., Polygône de Newton et b-fonctions pour LES modules microdifférentiels, Ann. sci. école norm. sup., 20, 391-441, (1987) · Zbl 0646.58021
[2] Sabbah, C., \(D\)-modules et cycles évanescents, (), 53-98
[3] Kashiwara, M.; Kawai, T., On holonomic systems of micro-differential equations III, Publ. res. inst. math. sci., 17, 813-979, (1981) · Zbl 0505.58033
[4] Laurent, Y.; Mebkhout, Z., Pentes algébriques et pentes analytiques d’un \(\mathcal{D}\)-module, Ann. sci. école norm. sup., 32, 39-69, (1999) · Zbl 0944.14007
[5] Assi, A.; Castro-Jiménez, F.J.; Granger, M., How to compute the slopes of a \(\mathcal{D}\)-module, Compos. math., 104, 107-123, (1996) · Zbl 0862.32005
[6] Castro-Jiménez, F.J.; Takayama, N., Singularities of the hypergeometric system associated with a monomial curve, Trans. amer. math. soc., 355, 3761-3775, (2003) · Zbl 1060.33023
[7] Gel’fand, I.M.; Zelevinsky, A.V.; Kapranov, M.M., Hypergeometric functions and toric varieties, Funct. anal. appl., 23, 94-106, (1989) · Zbl 0721.33006
[8] Granger, M.; Oaku, T.; Takayama, N., Tangent cone algorithm for homogenized differential operators, J. symbolic comput., 39, 417-431, (2005) · Zbl 1120.32300
[9] Oaku, T., An algorithm of computing \(b\)-functions, Duke math. J., 87, 115-132, (1997) · Zbl 0893.32009
[10] Oaku, T.; Takayama, N., Algorithms for \(D\)-modules — restriction, tensor product, localization, and local cohomology groups, J. pure appl. algebra, 156, 267-308, (2001) · Zbl 0983.13008
[11] N. Takayama, Kan/sm1, http://www.math.kobe-u.ac.jp/KAN/
[12] Schulze, M.; Walther, U., Irregularity of hypergeometric systems via slopes along coordinate subspaces, Duke math. J., 142, 465-509, (2008) · Zbl 1144.13012
[13] Adolphson, A., Hypergeometric functions and rings generated by monomials, Duke math. J., 73, 269-290, (1994) · Zbl 0804.33013
[14] Saito, M.; Sturmfels, B.; Takayama, N., Gröbner deformations of hypergeometric differential equations, (2000), Springer Berlin, Heidelberg · Zbl 0946.13021
[15] Kashiwara, M.; Oshima, T., Systems of differential equations with regular singularities and their boundary value problems, Ann. math., 106, 145-200, (1977) · Zbl 0358.35073
[16] Laurent, Y.; Monteiro Fernandes, T., Systèmes différentiels fuchsiens le long d’une sous-variété, Publ. res. inst. math. sci., 24, 397-431, (1988) · Zbl 0704.35032
[17] Baouendi, M.S.; Goulaouic, C., Cauchy problems with characteristic initial hypersurface, Comm. pure appl. math., 26, 455-475, (1973) · Zbl 0256.35050
[18] Greuel, G.M.; Pfister, G., A singular introduction to commutative algebra, (2002), Springer Berlin, Heidelberg
[19] Cox, D.; Little, J.; O’Shea, D., Ideals, varieties, and algorithms, (1991), Springer-Verlag New York
[20] Assi, A.; Castro-Jiménez, F.J.; Granger, M., The analytic standard Fan of a \(\mathcal{D}\)-module, J. pure appl. algebra, 164, 3-21, (2001) · Zbl 1001.32005
[21] Castro, F., Calcul effectifs pour LES idéaux d’opérateurs différentiels, (), 53-98
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.