Gröbner bases and logarithmic D -modules

Let C [ x ] = C [ x 1 , … , x n ] be the ring of polynomials with complex coefficients and A n the Weyl algebra of order n over C . Elements in A n are linear differential operators with polynomial coefficients. For each polynomial f , the ring M = C [ x ] f of rational functions with poles along f...

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Bibliographic Details
Published inJournal of symbolic computation Vol. 41; no. 3; pp. 317 - 335
Main Authors Castro-Jiménez, F.J., Ucha-Enríquez, J.M.
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.03.2006
Subjects
[formula omitted]-modules
Free divisors
Gröbner bases
Spencer divisors
Weyl algebra
Gröbner bases
Spencer divisors
D -modules
Weyl algebra
Free divisors
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Summary:Let C [ x ] = C [ x 1 , … , x n ] be the ring of polynomials with complex coefficients and A n the Weyl algebra of order n over C . Elements in A n are linear differential operators with polynomial coefficients. For each polynomial f , the ring M = C [ x ] f of rational functions with poles along f has a natural structure of a left A n -module which is finitely generated by a classical result of I.N. Bernstein. A central problem in this context is how to find a finite presentation of M starting from the input f . In this paper we use Gröbner base theory in the non-commutative frame of the ring A n to compare M to some other A n -modules arising in Singularity Theory as the so-called logarithmic A n -modules. We also show how the analytic case can be treated with computations in the Weyl algebra if the input data f is a polynomial.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2004.04.011