## Toshinori Oaku

### Research interests: Algebraic analysis

1. Algebraic study of systems of linear partial differential equations and generalized functions
2. Algorithms and programming for D-modules

### Programs

• bsi : a Risa/Asir program for computing local and global Bernstein-Sato ideals of polynomial mappings.
• annlog : a Risa/Asir program for computing the annihilator of f^s(\log f)^m for a polynomial f.

### Preprints

1. T. Oaku: Algorithms for D-modules, integration, and generalized functions with applications to statistics. to appear in Advanced Studies in Pure Mathematics PDF
2. T. Oaku: Localization, local cohomology, and the b-function of a D-module with respect to a polynomial. to appear in Advanced Studies in Pure Mathematics arXiv
3. T. Oaku: Operational calculus for holonomic distributions in the framework of D-module theory arXiv to appear in RIMS Kokyuroku Bessatsu, Kyoto Univ.
4. T. Oaku: Length and multiplicity of the local cohomology with support in a hyperplane arrangement. arXiv

### Publications

1. T. Oaku: Annihilators of Laurent coefficients of the complex power for normal crossing singularity. RIMS Kokyuroku Bessatsu B57 (2016), pp. 79--84. Preprint:arXiv
2. N. Marumo, T. Oaku, A. Takemura: Properties of powers of functions satisfying second-order linear differential equations with applications to statistics. Japan Journal of Industrial and Applied Mathematics: Volume 32, Issue 2 (2015), 553-572. Preprint: arXiv
3. T. Oaku: Annihilators of distributions associated with algebraic local cohomology of a hypersurface. Complex Variables and Elliptic Equations 59 (2014), 1533-1546.
4. T. Oaku: Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities. Journal of Symbolic Computation 50 (2013), 1--27. http://dx.doi.org/10.1016/j.jsc.2012.05.004. Preprint: arXiv:1108.4853v2[cs.SC]
5. T. Oaku: An algorithm to compute the differential equations for the logarithm of a polynomial. In Proceedings of the 37th International Conference on Symbolic and Algebraic Computation (ISSAC 2012), pages 273--280. ACM Press, 2012. Preprint: arXiv:1201.3198v1[cs.SC]
6. R. Bahloul, T. Oaku: Local Bernstein-Sato ideals: algorithm and examples. J. Symbolic Computation 45 (2010), 46-59. Preprint: arXiv:0806.4869
7. T. Oaku: Regular b-functions of D-modules. J. Pure and Applied Algebra 213 (2009), 1545-1557. PDF file
8. M. Granger, T. Oaku, N. Takayama: Tangent cone algorithm for homogenized differential operators. J. Symbolic Computation 39 (2005), 417-431. PDF file
9. M. Granger, T. Oaku: Minimal filtered free resolutions for analytic D-modules. J. Pure and Applied Algebra 191 (2004), 157-180. PDF file
10. T. Oaku, Y.Shiraki, N.Takayama: Algebraic algorithms for D-modules and numerical analysis. Computer mathematics (Proceedings of ASCM 2003), 23--39, Lecture Notes Ser. Comput., 10, World Sci. Publ., River Edge, NJ, 2003. PDF file .
11. T. Oaku: Minimal free resolutions of analytic D-modules. Proceedings of the 10th International Conference on Complex Analysis, Pusan, Korea, 2002, pp. 145-151. PDF file
12. T. Oaku, N. Takayama: Minimal free resolutions of homogenized D-modules. J. Symbolic Computation 32 (2001), 575-595.
13. T. Oaku, N. Takayama, H. Tsai: Polynomial and rational solutions of holonomic systems. J. Pure and Applied Algebra 164 (2001), 199-220. arXiv:math/0001064
14. T. Oaku, N. Takayama: Algorithms for D-modules --restriction, tensor product, localization, and local cohomology groups. J. Pure and Applied Algebra 156 (2001), 267-308. PDF file
15. T. Oaku, N. Takayama, U. Walther: A localization algorithm for D-modules. J. Symbolic Computation 29 (2000), 721-728. arXiv:math/9811030
16. T. Oaku, N. Takayama: An algorithm for de Rham cohomology groups of the complement of an affine variety. J. Pure and Applied Algebra 139 (1999), 201-233. . arXiv:math/9801114
17. T. Oaku, S. Yamazaki: Higher-codimensional boundary value problems and F-mild microfunctions --Local and microlocal uniqueness--. Publication of the Research Institute for Mathematical Sciences, Kyoto Univ. 34 (1998), 383-437 .
18. T. Oaku: Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules. Advances in Applied Math. 19 (1997), 61-105. PDF file Errata
19. T. Oaku: Algorithms for the b-function and D-modules associated with a polynomial. J. Pure Appl. Algebra 117 & 118 (1997), 495-518. PDF file
20. T. Oaku: An algorithm of computing b-functions. Duke Math. J. 87 (1997), 115-132. PDF file
21. T. Oaku: Groebner bases for D-modules on a non-singluar affine algebraic variety. Tohoku Math. J. 48 (1996), 575-600.
22. T. Oaku: Algorithmic methods for Fuchsian systems of linear partial differential equations. J. Math. Soc. Japan 47 (1995), 297-328 .
23. T. Oaku, T. Shimoyama: A Groebner basis method for modules over rings of differential operators. J. Symbolic Computation 18 (1994), 223-248.
24. T. Oaku: Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients. Japan J. Indust. Appl. Math. 11 (1994), 485-497.
25. T. Oaku: Boundary value problem for Fuchsian partial differential equations and reflection of singularities. Banach Center Publications 27 (1992), 337-348.
26. T. Oaku: Some remarks on local solvability of Fuchsian partial differential equations for hyperfunctions. Yokohama Math. J. 37 (1989), 139-143.
27. T. Oaku: Local solvability of Fuchsian elliptic equations for hyperfunctions. Yokohama Math. J. 36 (1988), 109-114.
28. T. Oaku: Higher codimensional boundary value problem and F-mild hyperfunctions. In: Algebraic Analysis, eds M. Kashiwara, T. Kawai, Academic Press, 1988, pp. 571-586.
29. T. Oaku: Boundary value problems for systems of linear partial differential equations and propagation of micro-analyticity. J. Fac. Sci. Univ. Tokyo 33 (1986), 175-232.
30. T. Oaku: Microlocal boundary value problem for Fuchsian operators, I. -F-mild microfunctions and uniqueness theorem. J. Fac. Sci. Univ. Tokyo 32 (1985), 287-317.
31. T. Oaku: F-mild hyperfunctions and Fuchsian partial differential equations. Advanced Studies in Pure Mathematics 4 (1984), 223-242.
32. T. Oaku: Analytic hypo-ellipticity and propagation of regularity for a system of microdifferential equations with non-involutory characteristics. Math. Ann. 269 (1984), 145-168.
33. T. Oaku: A canonical form of a system of microdifferential equations with non-involutory characteristics and branching of singularities. Inventiones Math. 65 (1982), 491-525.
34. T. Oaku: Micro-local Cauchy problems and local boundary value problems. Proc. Japan Acad. 55 (1979), 136-140.

### Slides of talks

1. T. Oaku: An algorithmic study on the intgration of holonomic distributions. New development of microlocal analysis and singular perturbation theory, RIMS, Kyoto University, October 3-7, 2016. T. Oaku: Some algorithmic problems for holonomic distributions. Microlocal Analysis and Singular Perturbation Theory, RIMS, Kyoto University, October 5-9, 2015.
2. T. Oaku: Algorithms for D-modules, integration, and generalized functions. The 8th Mathematical Society of Japan Seasonal Institute "Current Trends on Groebner Bases -The 50th Anniversary of Groebner Bases-", Hotel Nikko Osaka, Osaka, Japan, July 1-10, 2015. chapter 1 , chapter 2 , chapter 3 , chapter 4 , chapter 5
3. T. Oaku: Some D-module theoretic aspects of the local cohomology of a polynomial ring. The 8th Mathematical Society of Japan Seasonal Institute "Current Trends on Groebner Bases -The 50th Anniversary of Groebner Bases-", Hotel Nikko Osaka, Osaka, Japan, July 1-10, 2015.
4. T. Oaku: Algebraic and algorithmic study of some generalized functions associated with a real polynomial (or a real analytic function) Seminar at the department of algebra, the University of Sevilla, Spain, September, 2014.
5. T. Oaku: Algorithms for D-modules applied to generalized functions. Summer school "Algebra, Algorithms and Algebraic Analysis", Rolduc Abbey, the Netherlands, September 2-6, 2013.
6. T. Oaku: An algorithm to compute the differential equations for the logarithm of a polynomial. the 37th International Conference on Symbolic and Algebraic Computation (ISSAC 2012), Grenoble, July 2012.
e-mail: oaku at lab.twcu.ac.jp