The analytic continuation of generalized functions with respect to a parameter

“…Let The consideration of the solutions to the functional equation above was originally motivated by the problem of analytically continuing the zeta function attached to f, see [3,Theorem 1]. We would like to highlight two other types of consequences of that result.…”

Section: Bernstein-sato Polynomialmentioning

confidence: 99%

“…Namely it was the latter's first application and has been crucial in singling out the importance of holonomicity [3,Corollary (1.4.b)]. Moreover it is used to prove the stability of holonomicity under operations, e.g.…”

Section: Bernstein-sato Polynomialmentioning

confidence: 99%

On a theory of the b-function in positive characteristic

Bitoun

2018

Sel. Math. New Ser.

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We present a theory of the b-function (or Bernstein-Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic p > 0. Its b-function b f is defined to be an ideal of the algebra of continuous k-valued functions on Z p . The zero-locus of the b-function is thus naturally interpreted as a subset of Z p , which we call the set of roots of b f . We prove that b f has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Mustaţȃ and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of b f and relate it to the test ideals of f.

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“…By using these formulas, we have the following proposition, which is proved in a well-known way in the theory of analytic continuation of complex powers of polynomials. See for example Bernstein [1] …”

Section: We Say Thatmentioning

confidence: 99%

“…The real connected algebraic group G := GL n (R) + × SL n (R) acts on Mat n (R) algebraically by (1) (g, x) −→ g · x := g 1 x t g 2 with g := (g 1 , g 2 ) ∈ G and x ∈ V . Here, GL n (R) + := {g ∈ Mat n (R) | det(g) > 0} and SL n (R) := {g ∈ Mat n (R) | det(g) = 1}.…”

Section: §1 Introductionmentioning

confidence: 99%

Singular invariant hyperfunctions on the square matrix space and the alternating matrix space

Muro

2003

Nagoya Mathematical Journal

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Abstract. Fundamental calculations on singular invariant hyperfunctions on the n × n square matrix space and on the 2n × 2n alternating matrix space are considered in this paper. By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctions as their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasirelatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions. §1. Introduction Let V := Mat n (R) be the real vector space of n × n real matrices. The real connected algebraic group G := GL n (R) + × SL n (R) acts on Mat n (R) algebraically by. Then the pair (G, V ) is a regular prehomogeneous vector space with an irreducible relative invariant P (x) := det(x) and V decomposes into a finite number of G-orbits. This is the first object of this paper. Next, let V := Alt 2n (R) be the real vector space of 2n × 2n real alternating matrices. The real connected algebraic group G := GL 2n (R) + acts on Alt 2n (R) algebraically by (2) (g, x) −→ g · x := gx t g with g ∈ G and x ∈ V . Then the pair (G, V ) is a regular prehomogeneous vector space with an irreducible relative invariant P (x) := Pf(x) and V decomposes into a finite number of G-orbits. Here Pf(x) stands for the Pfaffian of the alternating matrix x which is a polynomial on Alt 2n (R) given as a square root of det(x) taking the value Pf(J n ) = (−1) n(n−1)/2 for J n := 0n In −In 0n . This is the second object in this paper. These two prehomogeneous vector spaces have some common properties. Both of two have two open G-orbits, one is V + := {x ∈ V | P (x) > 0} and the other is V − := {x ∈ V | P (x) < 0}. We define the complex power of the relative invariant P (x) byfor s ∈ C. Then it is well known that the integral |P (x)| s ± f (x) dx is absolutely convergent for all the rapidly decreasing function f (x) provided that the real part (s) of s ∈ C is non-negative. In addition, |P (x)| s ± can be regarded as a tempered distribution -and hence a hyperfunctionon V with a holomorphic parameter s ∈ C and it is extended to the whole complex plane as a meromorphic function in s ∈ C. The poles of P [ a,s] (x) belong to the set Z <0 := {−1, −2, . . . }. We define the linear combination offor a := (a + , a − ) ∈ C 2 and s ∈ C, which is a relatively invariant hyperfunction under the action of G. Namely we havefor all g ∈ G with χ(g) = det(g 1 ) when V = Mat n (R) or with χ(g) = det(g) when V = Alt 2n (R). For a fixed complex number λ ∈ C, we call a...

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The analytic continuation of generalized functions with respect to a parameter

Cited by 234 publications

References 4 publications

Gelfand-Kirillov dimension of commutative subalgebras of simple inﬁnite dimensional algebras and their quotient division algebras

Gelfand-Kirillov dimension of commutative subalgebras of simple inﬁnite dimensional algebras and their quotient division algebras

On a theory of the b-function in positive characteristic

Singular invariant hyperfunctions on the square matrix space and the alternating matrix space

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