Bernsteinsato polynomials and analytic continuation of f s. The name bfunction comes from this theory and the socalled a,b,c functions of m. We survey the theory of poisson traces or zeroth poisson homology developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of singular poisson varieties, conditions for it to be finitedimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. Assume that g is connected and reductive, and that v has an open gorbit. Choose a nonzero relative invariant fv in cvv, with character. A rich source of polynomials satisfying local functional equations is the theory of prehomogeneous vector spaces.
Jul 22, 2012 elements of computeralgebraic analysis tutorial abstract viktor levandovskyy lehrstuhl d fur mathematik rwth aachen templergraben 64 52062 aachen, germany viktor. As a corollary, we show that there exists an integer t depending only on. O is the monic generator of the ideal of the polynomials bs. The bernstein sato polynomial or global bfunction is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of dmodules. The two seminal papers on this subject are certainly kashiwaras thesis december 70 and bernsteins papers of 71 and 72 published in functional analysis 3, 4. In mathematics, a dmodule is a module over a ring d of differential operators.
Intersection homology dmodule 647 let us recall that the bernstein polynomial b fs of a nonzero germ f. Symbolic computation 2001 32, 643662 algorithm for computing bernsteinsato. In the proof above, the identity 8f 8f did is derived from the identity f. Intersection homology dmodule and bernstein polynomials. Local functional equations of polarizations 311 we obtain rx f. For instance, the roots of the bernstein sato polynomial are related to the jumping.
Algebraic study of systems of partial differential equations. Bernsteinsato polynomials and functional equations submitted by emmanuel lemoine on thu, 120520 15. The two seminal papers on this subject are certainly kashiwaras thesis december 70 and bernstein s papers of 71 and 72 published in functional analysis 3, 4. Algebraic and algorithmic study of some generalized functions. The idea is to replace z p by ct in this case fis a polynomial with complex coe cients. We show that for xed n and d the set of bernsteinsato polynomials of all the polynomials in at most n variables of degrees at most d is nite. Bernsteinsato polynomial associated to an algebraic variety. It is observed that this polynomial can have zero as a root, or even positive roots. The existence of functional equations of the bernsteinsato type for the products f fp\p was proved simultaneously by c. As its name indicates, a dmodule is a module over the sheaf of rings d of. Pdf on bernsteinsato ideals and decomposition of d.
The minimal such polynomial is known as the bernstein sato polynomial, and its roots determine the arithmetic progressions mentioned above. Functional equations in limoges felim 2011 constructive. After providing a survey of known algorithms for computing the global b function, we develop a new method to compute the local b function for a single polynomial. An algebraic proof by mebkhout and narvaez can be found in 27. After providing a survey of known algorithms for computing the global bfunction, we develop a new method to compute the local bfunction for a single polynomial. Pdf a bernstein polynomial approach for solution of. Bernsteinsato polynomials and spectral numbers request pdf. The unique monic polynomial of smallest degree satisfying this functional equation is called the bernstein sato polynomial of f. Quasihomogeneous and semiquasihomogeneous isolated singularities. On the existence of a functional equation in the polynomial case. This polynomial was simultaneously introduced by mikio sato in a di erent context in view of giving functional equations for relative invariants of prehomogeneous spaces and of studying zeta functions associated with them, see 32, 33. Ag 24 jul 2019 bernsteinsato functional equations, vfiltrations, and multiplier ideals of direct summands josep alvarez montaner1, daniel j. He gave a series of talks on these topics at tokyo university but had to stop for lack of combatants.
We also make an extensive study of an example for which we give an expression of a generic and under some conditions, a relative bernsteinsato polynomial. In this part, we give constructive results generalizing previous ones obtained by the author in the case of one function. Kx, the global bernsteinsato polynomial of f, denoted bf, is the monic polynomial bs. Then, as0 is the skew polynomial ring of forward resp. Poisson traces, dmodules, and symplectic resolutions. Dmodules, bernsteinsato polynomials and finvariants of. Functional equations in limoges felim 2011 constructive algebra for systems theory march1416,2011 organisedby moulaybarkatou,thomascluzeau,caroleelbacha andjacquesarthurweil takingplaceat lasalleder. Therefore the existence of a rational function f satisfying f. Zeta functions in algebraic geometry mircea mustat. In particular, we use the algebraic analysis approach to linear systems theory to study builtin properties of linear functional systems. The existence of the bernsteinsato polynomial for a direct summand of a polynomial over a field is proved in this context. Let f in cv be a nonzero relative invariant with character. In order to avoid cumbersome studying of the linear system of equations to derive the poperators, the papers 8, 9, 10 and 11, proposed to apply the functional relation. The focus of this work lies on the relation between bernsteinsato ideals and the decomposition of the d2module m.
General terms algorithms, theory tion, we will provide results on gelfandkirillov dimension, generalized krull dimension as well as global homological dimension. In the mathematical field of numerical analysis, a bernstein polynomial, named after sergei natanovich bernstein, is a polynomial in the bernstein form, that is a linear combination of bernstein basis polynomials. Bernsteinsato polynomials and functional equations m granger differential algebraic groups b malgrange. This is an important invariant in the theory of singularities see. Since around 1970, dmodule theory has been built up, mainly as a response to the ideas of mikio sato on algebraic analysis, and expanding on the work of sato and joseph bernstein on the bernsteinsato. Bernsteinsato polynomials of ideals and varieties8 2. In this paper, we focus on linear functional systems, i. We give a polynomial formula for these invariants depending on a suitable congruence of p and show that all the roots of the bernstein sato polynomial can be obtained by our method. After a convenient shifting, b i s is also an invariant of the variety v i. Using standard basis techniques, our aim is to present an algorithm for computing generators of b j and b. For instance, the roots of the bernsteinsato polynomial are related to the jumping. Since its inception, the bernsteinsato polynomial has found broad applications in the study of singularities. An overview 5 igusas zeta function also has a motivic version, due to denef and loeser, see dl. We survey some applications to generalized bernsteinsato polynomials, multiplier ideals, and monodromy of vanishing cycles.
Algebraic analysis has been developed by malgrange, bernstein, sato, kashiwara. The polynomial bfs is called the bernsteinsato polynomial of f and is always a. The bernsteinsato polynomial for the variety f1 fp 0 budurmustatasaito 2006 can be computed as the bfunction or the indicial polynomial with a shift of this holonomic system along the submanifold t1 tp 0. Bernsteinsato functional equations 3 bernsteinsato polynomials are closely related to the notion of a v. We then develop algorithms that compute generalized. For the bernstein polynomial in dmodule theory, see bernsteinsato polynomial. In this mostly expository note we give a downtoearth introduction to the vfiltration of m. Proof of the existence of the bfunction for a an analytic germ. An overview zeta functions encode the counting of certain objects of geometric, algebraic, or. Algorithms for bernsteinsato polynomials and multiplier. If our goal is to study linear systems of differential timedelay dtd equations, of differential difference equations, of partial differential equations and so on, i. Algorithm for computing bernsteinsato ideals associated. On bernsteinsato polynomials gennady lyubeznik communicated by ken goodearl abstract.
The major interest of such dmodules is as an approach to the theory of linear partial differential equations. Convolution dirichlet series and a kronecker limit formula for secondorder eisenstein series jorgenson, jay and osullivan, cormac. The existence of the bernstein sato polynomial for a direct summand of a polynomial over a field is proved in this context. Since its inception, the bernstein sato polynomial has found broad applications in the study of singularities. Bernsteinsato polynomials and functional equations. However, it only appeared in the 70s for systems of partial differential equations with analytic coefficients, after pioneering talks by sato in the 60s and quillen thesis in 64. Terras, functional equations of generalized epstein zeta functions in several variables, nagoy math. After providing a survey of known algorithms for computing the.
Let us calculate the bernsteinsato polynomial of the zero polynomial. The unique monic polynomial of smallest degree satisfying this functional equation is called the bernsteinsato polynomial of f. The bernsteinsato polynomial can also be defined for products of powers of several polynomials sabbah 1987. Bernstein sato polynomials and analytic continuation of f s. We also make an extensive study of an example for which we give an expression of a generic and under some conditions, a relative bernstein sato polynomial. The minimal such polynomial is known as the bernsteinsato polynomial, and its roots determine the arithmetic progressions mentioned above. The polynomial bfs is called the bernsteinsato polynomial of. So the strategy from localisation is to use the information from the localised situation. Our main new result is the description of different types of bernsteinsato ideals of. The existence of functional equations of the bernstein sato type for the products f fp\p was proved simultaneously by c. Bernsteinsato polynomials and functional equations request pdf.
Moreover, they showed that the bernsteinsato polynomial of f considered as an element of the direct. Algebraic and algorithmic study of some generalized. A prehomogeneous vector space is a vector space v over a eld kof charac. Similarly, smalls book 38 is a very enjoyable, well written book and focuses on the most essential aspects of functional equations. We give a polynomial formula for these invariants depending on a suitable congruence of p and show that all the roots of the bernsteinsato polynomial can be obtained by our method. Again, the model is provided by the riemann zeta function.
Now, let us determine the bernsteinsato polynomial for. Pdf on bernsteinsato ideals and decomposition of dmodules. In this case it is a product of linear factors with rational coefficients. While the main goal is classi cation, functional relations such as 1. Let g be a complex linear algebraic group and g glv a finite dimensional rational representation.
His ideas were reconsidered and systematically developed by masaki kashiwara in his 1969 thesis 1, 2. Toshinori oaku tokyo womans christian universityalgorithms for dmodules applied to generalized functions. We associate with these polynomials a number of functional equations and related ideals b, b j and b. Given a collection of germs of holomorphic functions f. The polynomials b fs are called bernsteinsato polynomials in order to take this double origin into account. Almost all known examples of local functional equations are of this.
Example 1 let a 1 be the polynomial and b 1 the rational weyl algebra. As mentioned above, according to its definition definition 1. Intersection homology dmodules and bernstein polynomials. Algorithms for dmodules applied to generalized functions. Character sums asociated to prehomogeneous vector spaces. Vfiltrations is an essential tool in the theory of dmodules over regular rings containing a field of characteristic zero. Since around 1970, dmodule theory has been built up, mainly as a response to the ideas of mikio sato on algebraic analysis, and expanding on the work of sato and joseph bernstein on the bernsteinsato polynomial. Algorithm for computing bernsteinsato ideals associated with.