scholarly journals THE HOLOMORPHY CONJECTURE FOR NONDEGENERATE SURFACE SINGULARITIES

2016 ◽  
Vol 227 ◽  
pp. 160-188
Author(s):  
WOUTER CASTRYCK ◽  
DENIS IBADULA ◽  
ANN LEMAHIEU
Keyword(s):  
Zeta Function ◽  
Arbitrary Dimension ◽  
Newton Polyhedron ◽  
Character Sums ◽  
Specific Character ◽  
Trivial Character ◽  
Surface Singularities ◽  
Local Monodromy

The holomorphy conjecture roughly states that Igusa’s zeta function associated to a hypersurface and a character is holomorphic on$\mathbb{C}$whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over$\mathbb{C}$with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by$B_{1}$-facets.

scholarly journals THE BOUNDARY LERCH ZETA-FUNCTION AND SHORT CHARACTER SUMS À LA Y. YAMAMOTO

2020 ◽  
Vol 74 (2) ◽  
pp. 313-335
Author(s):  
Xiaohan WANG ◽  
Jay MEHTA ◽  
Shigeru KANEMITSU
Keyword(s):  
Zeta Function ◽  
Character Sums ◽  
Lerch Zeta Function

scholarly journals Motivic invariants of real polynomial functions and their Newton polyhedrons

2015 ◽  
Vol 160 (1) ◽  
pp. 141-166 ◽  
Author(s):  
GOULWEN FICHOU ◽  
TOSHIZUMI FUKUI
Keyword(s):  
Zeta Function ◽  
Polynomial Function ◽  
Newton Polyhedron ◽  
Polynomial Functions ◽  
Homogeneous Polynomials ◽  
Real Polynomial ◽  
Motivic Zeta Function

AbstractWe give an expression of the motivic zeta function for a real polynomial function in terms of the Newton polyhedron of the function. As a consequence, we show that the weights are determined by the motivic zeta function for convenient weighted homogeneous polynomials in three variables. We apply this result to the blow-Nash equivalence.

scholarly journals Local zeta functions and Newton polyhedra

2003 ◽  
Vol 172 ◽  
pp. 31-58 ◽  
Author(s):  
W. A. Zuniga-Galindo
Keyword(s):  
Zeta Function ◽  
Valuation Ring ◽  
Zeta Functions ◽  
Newton Polyhedron ◽  
Group Of Units ◽  
Local Zeta Function ◽  
Degenerate Polynomial ◽  
Small Set ◽  
Local Zeta Functions ◽  
Almost All

AbstractTo a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.

scholarly journals Geometric determination of the Poles of Highest and second Highest order of Hodge And motivic Zeta functions

2004 ◽  
Vol 176 ◽  
pp. 1-18
Author(s):  
B. Rodrigues
Keyword(s):  
Zeta Function ◽  
Arbitrary Dimension ◽  
Zeta Functions ◽  
Motivic Zeta Functions ◽  
Hodge Polynomials ◽  
Motivic Zeta Function

AbstractTo any f ∈ ℂ[x1, … ,xn] \ ℂ with f(0) = 0 one can associate the motivic zeta function. Another interesting singularity invariant of f-1{0} is the zeta function on the level of Hodge polynomials, which is actually just a specialization of the motivic one. In this paper we generalize for the Hodge zeta function the result of Veys which provided for n = 2 a complete geometric determination of the poles. More precisely we give in arbitrary dimension a complete geometric determination of the poles of order n − 1 and n. We also show how to obtain the same results for the motivic zeta function.

scholarly journals MACDONALD INTEGRALS AND MONODROMY

2001 ◽  
Vol 12 (09) ◽  
pp. 987-1004
Author(s):  
JAN DENEF ◽  
FRANÇOIS LOESER
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Coxeter Group ◽  
Finite Coxeter Group ◽  
Local Monodromy

We prove several results on monodromies associated to Macdonald integrals, that were used in our previous work on the finite field analogue of a conjecture of Macdonald. We also give a new proof of our formula expressing recursively the zeta function of the local monodromy at the origin of the discriminant of a finite Coxeter group in terms of the degrees of the group.

scholarly journals Monomial deformations of certain hypersurfaces and two hypergeometric functions

2015 ◽  
Vol 11 (08) ◽  
pp. 2405-2430 ◽  
Author(s):  
Kazuaki Miyatani
Keyword(s):  
Zeta Function ◽  
Finite Fields ◽  
Hypergeometric Functions ◽  
Zeta Functions ◽  
Character Sums ◽  
Rational Points ◽  
Explicit Description

The purpose of this paper is to give an explicit description, in terms of hypergeometric functions over finite fields, of the zeta functions of certain smooth hypersurfaces that generalize the Dwork family. The key point here is that we count the number of rational points employing both the techniques of character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.

Certain Subclasses of Bi-Univalent Functions Involving Double Zeta Function

2015 ◽  
Vol 4 (4) ◽  
pp. 28-33
Author(s):  
Dr. T. Ram Reddy ◽  
◽  
R. Bharavi Sharma ◽  
K. Rajya Lakshmi ◽  
◽  
...  
Keyword(s):  
Zeta Function ◽  
Univalent Functions ◽  
Double Zeta
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