Algorithms for \(b\)-functions, induced systems, and algebraic local cohomology of \(D\)-modules
Let \(K\) be a field of characteristic \(0\) and denote by \(A_{n+1}(K)\) the Weyl algebra attached to the polynomial ring \(K[t,x_1,\dots,x_n]\). The author gives algorithms for detecting whether a finitely generated module over \(A_{n+1}(K)\) is specializable along \(t=0\) and for computing the corresponding \(b\)-functions. He also obtains algorithms for computing restrictions and algebraic local cohomology with respect to \(t=0\). The main ingredient is the technique of Gröbner bases for rings of differential operators, as introduced in [\textit{J. Briançon} and \textit{Ph. Maisonobe}, Enseign. Math. 30, 7-38 (1984; Zbl 0542.14008)] and [\textit{F. Castro}, C. R. Acad. Sci., Paris, Sér. I 302, No. 14, 487-490 (1986; Zbl 0606.32007)]. In order to compute Gröbner bases with respect to the Malgrange-Kashiwara filtration, the author uses a homogenization process for elements in \(A_{n+1}(K)\), which is analogous to the algorithm for the tangent cone in local algebra. A similar idea has been used in [\textit{A. Assi, F. J. Castro-Jiménez} and \textit{J. M. Granger}, C. R. Acad. Sci., Paris, Sér. I 320, No. 2, 193-198 (1995; Zbl 0849.13019) and Compos. Math. 104, No. 2, 107-123 (1996; Zbl 0862.32005)] [see also \textit{F. J. Castro-Jiménez} and the reviewer, Homogenising differential operators, Prep. 36, Fac. Matemáticas, Univ. Sevilla, June (1997)]. Full details have appeared in the author's paper [Duke Math. J. 87, No. 1, 115-132 (1997; see the paper above)].
scientific article; zbMATH DE number 993541
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