A survey of Igusa’s local zeta function

Meuser, Diane

doi:10.1353/ajm.2016.0006

Cited by 20 publications

(17 citation statements)

References 64 publications

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“…The basic tool he used is called the π-adic stationary phase formula, which was first introduced by Igusa [8] and then became a very powerful tool in the study of the rationality of Z f (s, χ) in arbitrary characteristic case. For more progress in the rationality of Igusa's local zeta function, see [11].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Igusa local zeta functions of a class of hybrid polynomials

Yin,

Hong

2016

Preprint

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In this paper, we study the Igusa's local zeta functions of a class of hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. Such class of hybrid polynomial was first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We prove the rationality of these local zeta functions and describe explicitly their poles. The proof is based on Igusa's stationary phase formula.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Igusa local zeta functions of a class of hybrid polynomials

Yin,

Hong

2016

Preprint

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“…In [28], we showed that the p-adic open string N −point tree amplitudes are bonafide integrals that admit meromorphic continuations as rational functions, by relating them with multivariate local zeta functions (also called multivariate Igusa local zeta functions [29,30]). Moreover Denef and Loeser [31] established that the limit p approaches to one of a local zeta function gives rise a new object called topological zeta function, which is associated with a complex polynomial.…”

Section: Jhep08(2018)043mentioning

confidence: 99%

“…The functions Z (N ) (s; I, 0, K e ) and Z (N ) (s; T I, 1, K e ) are multivariate local zeta functions, see Apendix B. The local zeta functions are related with deep arithmetical and geometrical matters, and they have been studied extensively since the 50s, see [30,44] and references therein.…”

Section: Non-archimedean String Zeta Functionsmentioning

confidence: 99%

On p-adic string amplitudes in the limit p approaches to one

Bocardo-Gaspar

García‐Compeán

Zúñiga–Galindo

2018

J. High Energ. Phys.

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In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa's local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa's local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser's theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

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“…Determining the convergence of the amplitudes in momentum space is a difficult task, both in the standard and p-adic case; however, for the latter, this was precisely done for the N-point tree amplitudes in [8]. In this article we show (in a rigorous mathematical way) that the p-adic open string N-point tree amplitudes are bona fide integrals that admit meromorphic continuations as rational functions, this is done by associating to them multivariate local zeta functions (also called multivariate Igusa local zeta functions) [25][26][27][28]. In [6] we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero (for instance R, C, Q p ) are bona fide integrals that admit meromorphic continuations in the kinematic parameters.…”

Section: Introductionmentioning

confidence: 99%

Local Zeta Functions and Koba–Nielsen String Amplitudes

et al. 2021

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This article is a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions (in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p→1. Gerasimov and Shatashvili studied the limit p→1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

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A survey of Igusa’s local zeta function

Cited by 20 publications

References 64 publications

Igusa local zeta functions of a class of hybrid polynomials

Igusa local zeta functions of a class of hybrid polynomials

On p-adic string amplitudes in the limit p approaches to one

Local Zeta Functions and Koba–Nielsen String Amplitudes

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