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Algorithms for \(b\)-functions, restrictions, and algebraic local cohomology groups of \(D\)-modules. (English) Zbl 0938.32005
From the introduction: “Our purpose is to present algorithms for computing some invariants and functors attached to algebraic \(D\)-modules by using Gröbner bases for differential operators. Let \(K\) be an algebraically closed field of characteristic zero and let \(X\) be a Zariski open set of \(K^n\) with a positive integer \(n\). We fix a coordinate system \(x=(x_1, \dots, x_n)\) of \(X\) and write \(\partial= (\partial_1, \dots, \partial_n)\) with \(\partial_i:= \partial/ \partial x_i\). We denote by \({\mathcal D}_X\) the sheaf of algebraic linear differential operators on \(X\).
Let \({\mathcal M}\) be a coherent left \({\mathcal D}_X\)-module and \(u\) a section of \({\mathcal M}\). Suppose that \(f=f(x) \in K[x]\) is an arbitrary nonconstant polynomial of \(n\) variables. If \({\mathcal M}\) is holonomic, then for each point \(p\) of \(Y:=\{x\in X|f(x)=0\}\), there exists a germ \(P(x,\partial,s)\) of \({\mathcal D}_X[s]\) at \(p\) and a polynomial \(b(s) \in K[s]\) of one variable so that \[ P(x,\partial,s)(f^{s+1}u)= b(s)f^su\tag{1.1} \] holds with an indeterminate \(s\). More precisely, (1.1) means that there exists a nonnegative integer \(m\) so that \[ Q:=f^{m-s}\bigl(b(s)-P(s,\partial, s) f \bigr)f^s\in {\mathcal D}_X[s] \] satisfies \(Qu=0\) in \({\mathcal M}[s]:=K[s] \otimes_K {\mathcal M}\). The monic polynomial \(b(s)\) of the least degree that satisfies (1.1), if any, is called the (generalized) \(b\)-function for \(f\) and \(u\) at \(p\). Suppose that a presentation (i.e., generators and the relations among them) of a coherent left \({\mathcal D}_X\)-module \({\mathcal M}\) and a section \(u\) of \({\mathcal M}\) are given. Then we are concerned with algorithms for solving the following problems:
(A1) to determine whether there exists and to find, if it does, the \(b\)-function for \(f\) and \(u\);
(A2) to obtain presentations of the algebraic local cohomology groups \({\mathcal H}^j_{[Y]} ({\mathcal M})\) \((j=0,1)\) as left \({\mathcal D}_X\)-modules;
(A3) to obtain a presentation of the localization \({\mathcal M}(*Y)= {\mathcal M}[f^{-1}]\) of \({\mathcal M}\) by \(f\) as a left \({\mathcal D}_X\)-module;
(A4) to obtain a presentation of the left \({\mathcal D}_X[s]\)-module \(\sum_{i= 1}^r {\mathcal D}_X[s] (f^s\otimes u_i)\), where \(u_1,\dots,u_r\) are generators of \({\mathcal M}\) and \(f^s\otimes u_i\) is regarded as a section of \(({\mathcal O}_X [s,f^{-1}]f^s) \otimes_{{\mathcal O}_X}{\mathcal M}\)”.

MSC:
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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