On Decomposition of D-modules and Bernstein-Sato polynomials for Hyperplane Arrangements

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Addis Ababa University


. This thesis discusses the relationship between Bernstein-Sato ideals of = xy(a3x + y):::(amx + y); ai 2 C; ai 6= aj ;m _ 3 and the decomposition of the D2-module M_ Chx; y; @x; @yi___ over the Weyl algebra Chx; y; @x; @yi, where for each i 2 f1; 2; :::;mg, m ; _i 2 C and _1 := x; _2 = y; _i := aix + y; (3 _ i _ m) are linear forms on C2. The thesis starts by summarizing the de_nition, properties and the results on the number of decomposition factors of M_ Then it continues with the de_nition and basic properties of univariate Bernstein-Sato polynomials, and collects what is known of Bernstein-Sato polynomials for hyperplane arrangements. A variation of the idea are the multivariate Bernstein-Sato polynomials and ideals. Main new results in the thesis are on the description of di_erent types of Bernstein- Sato ideals of = xy Qm i=3(aix + y) (in chapter 4) and on the use of these ideals in the decomposition of the D2-module M_ (in chapter



On Decomposition of D-modules, and Bernstein-Sato