Algorithms for the b-function and D-modules associated with a polynomial

Oaku, Toshinori

doi:10.1016/s0022-4049(97)00024-8

Cited by 42 publications

(48 citation statements)

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“…A satisfactory interpretation of all roots of b f (s) for general f , however, remains outstanding. Indeed, until [Oak96] there was not even an algorithm for the computation of b f (s) for an arbitrary polynomial f .…”

Section: Introductionmentioning

confidence: 99%

Bernstein–Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements

Walther¹

2004

Compositio Math.

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Let Q ∈ C[x 1 , . . . , x n ] be a homogeneous polynomial of degree k > 0. We establish a connection between the Bernstein-Sato polynomial b Q (s) and the degrees of the generators for the top cohomology of the associated Milnor fiber. In particular, the integerThe link is provided by the relative de Rham complex and D-module algorithms for computing integration functors.As an application we determine the Bernstein-Sato polynomial b Q (s) of a generic central arrangement Q = k i=1 H i of hyperplanes. In turn, we obtain information about the cohomology of the Milnor fiber of such arrangements related to results of Orlik and Randell who investigated the monodromy.We also introduce certain subschemes of the arrangement determined by the roots of b Q (s). They appear to correspond to iterated singular loci.

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Section: Introductionmentioning

confidence: 99%

Bernstein–Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements

Walther¹

2004

Compositio Math.

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“…, have been designed [60,76] and implemented [56], but it is still a highly nontrivial task both from theoretical and implementational point of view. Note that in the ∂-finite setting it is rather trivial to get the annihilator of f s , since it is hyperexponential in all the x i .…”

Section: Resultsmentioning

confidence: 99%

Advanced applications of the holonomic systems approach

Koutschan

2010

ACM Commun. Comput. Algebra

107

174

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Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Linz, September 2009Christoph Koutschan AbstractThe holonomic systems approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for the algorithmic treatment of holonomic function identities. Frédéric Chyzak later extended this framework by introducing the closely related notion of ∂-finite functions and by placing their manipulation on solid algorithmic grounds. For practical purposes it is convenient to take advantage of both concepts which is not too much of a restriction: The class of functions that are holonomic and ∂-finite contains many elementary functions (such as rational functions, algebraic functions, logarithms, exponentials, sine function, etc.) as well as a multitude of special functions (like classical orthogonal polynomials, elliptic integrals, Airy, Bessel, and Kelvin functions, etc.). In short, it is composed of functions that can be characterized by sufficiently many partial differential and difference equations, both linear and with polynomial coefficients. An important ingredient is the ability to execute closure properties algorithmically, for example addition, multiplication, and certain substitutions. But the central technique is called creative telescoping which allows to deal with summation and integration problems in a completely automatized fashion. Part of this thesis is our Mathematica package HolonomicFunctions in which the above mentioned algorithms are implemented, including more basic functionality such as noncommutative operator algebras, the computation of Gröbner bases in them, and finding rational solutions of parameterized systems of linear differential or difference equations.Besides standard applications like proving special function identities, the focus of this thesis is on three advanced applications that are interesting in their own right as well as for their computational challenge. First, we contributed to translating Takayama's algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessel's lattice path conjecture. The computations that completed the proof were of a nontrivial size and have been performed with our software. Second, investigating basis functions in finite element methods, we were able to extend the existing algorithms in a way that allowed us to derive various relations which generated a considerable speed-up in the subsequent numerical simulations, in this case of the propagation of electromagnetic waves. The third application concerns a computer proof of the enumeration formula for totally symmetric plane partitions, also known as Stembridge's theorem. To make the underlying computations feasible we employed a new approach for finding creative telescoping operators. KurzzusammenfassungIn den frühen 1990er Jahren beschr...

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“…This is in essence Theorem 18 of [33]. (See the remarks in Subsection 2.2 on how to compute such Gröbner bases.…”

Section: If We Identifymentioning

confidence: 93%

“…Section 3. Algorithms for the computation of these objects have been established by T. Oaku in a sequence of papers [31][32][33].…”

Section: The Master Planmentioning

confidence: 99%

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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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Algorithms for the b-function and D-modules associated with a polynomial

Cited by 42 publications

References 15 publications

Bernstein–Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements

Bernstein–Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements

Advanced applications of the holonomic systems approach

D-modules and Cohomology of Varieties

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