<i>D</i>-modules: an overview towards effectivity

Maisonobe, Ph.

doi:10.1017/cbo9780511565816.003

Cited by 8 publications

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“…One of the earliest such approaches are in [34,4], but at least implicitly Buchberger's algorithm in a Weyl algebra was discussed as early as [70]. An algorithmic approach to the isolated singularities case [144] preceded the general algorithms based on Gröbner bases in a non-commutative setting outlined below. 10.1.…”

Section: Appendix: Computability (By a Leykin)mentioning

confidence: 99%

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

Walther

2016

Invent. math.

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Abstract. For a germ f on a complex manifold X, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the D-module generated by f s to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of f s is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials.In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of 1/f ; we confirm in many cases a corresponding conjecture on the annihilator of f s but we disprove it in general; we show that the Bernstein-Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of f s is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.

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Section: Appendix: Computability (By a Leykin)mentioning

confidence: 99%

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

Walther

2016

Invent. math.

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“…In such a field, the coefficients are not necessarily constants ∈ K, but belongs to a ring of functions. We refer to [17], [16], for more information on this topic.…”

Section: Notationsmentioning

confidence: 99%

“…Proof. If T d is a bound for the coefficients in U k and A k at the step d, according to (13), (16), the size of the coefficients at the step d+1 is bounded by…”

Section: Complexitymentioning

confidence: 99%

Isolated points, duality and residues

Mourrain

1997

Journal of Pure and Applied Algebra

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In this paper, we are interested in the use of duality in effective computations on polynomials. We represent the elements of the dual of the algebra R of polynomials over the field K as formal series ∈ K[[∂]] in differential operators. We use the correspondence between ideals of R and vector spaces of K[[∂]], stable by derivation and closed for the (∂)-adic topology, in order to construct the local inverse system of an isolated point. We propose an algorithm, which computes the orthogonal D of the primary component of this isolated point, by integration of polynomials in the dual space K[∂], with good complexity bounds. Then we apply this algorithm to the computation of local residues, the analysis of real branches of a locally complete intersection curve, the computation of resultants of homogeneous polynomials.

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D-modules and Cohomology of Varieties

Walther

2002

Computations in Algebraic Geometry With Macaulay 2

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In this chapter we introduce the reader to some ideas from the world of differential operators. We show how to use these concepts in conjunction with Macaulay 2 to obtain new information about polynomials and their algebraic varieties.Gröbner bases over polynomial rings have been used for many years in computational algebra, and the other chapters in this book bear witness to this fact. In the mid-eighties some important steps were made in the theory of Gröbner bases in non-commutative rings, notably in rings of differential operators. This chapter is about some of the applications of this theory to problems in commutative algebra and algebraic geometry.Our interest in rings of differential operators and D-modules stems from the fact that some very interesting objects in algebraic geometry and commutative algebra have a finite module structure over an appropriate ring of differential operators. The prime example is the ring of regular functions on the complement of an affine hypersurface. A more general object is theČech complex associated to a set of polynomials, and its cohomology, the local cohomology modules of the variety defined by the vanishing of the polynomials. More advanced topics are restriction functors and de Rham cohomology.With these goals in mind, we shall study applications of Gröbner bases theory in the simplest ring of differential operators, the Weyl algebra, and develop algorithms that compute various invariants associated to a polynomial f . These include the Bernstein-Sato polynomial b f (s), the set of differential operators J(f s ) which annihilate the germ of the function f s (where s is a new variable), and the ring of regular functions on the complement of the variety of f .For a family f 1 , . . . , f r of polynomials we study the associatedČech complex as a complex in the category of modules over the Weyl algebra. The algorithms are illustrated with examples. We also give an indication what other invariants associated to polynomials or varieties are known to be computable at this point and list some open problems in the area.

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D-modules: an overview towards effectivity

Cited by 8 publications

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The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

The Jacobian module, the Milnor fiber, and the D-module generated by $$f^s$$ f s

Isolated points, duality and residues

D-modules and Cohomology of Varieties

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