scholarly journals On the Geometric Determination of the Poles of Hodge and Motivic Zeta Functions

2005 ◽  
Vol 2005 (578) ◽  
Author(s):  
B. Rodrigues
Keyword(s):  
Zeta Functions ◽  
Motivic Zeta Functions

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 EQUIVARIANT ZETA FUNCTIONS FOR INVARIANT NASH GERMS

2016 ◽  
Vol 222 (1) ◽  
pp. 100-136 ◽  
Author(s):  
FABIEN PRIZIAC
Keyword(s):  
Finite Group ◽  
Zeta Functions ◽  
Poincaré Series ◽  
Linear Action ◽  
Poincare Series ◽  
Motivic Zeta Functions ◽  
A Finite Group

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef–Loeser formulas for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

scholarly journals Higher-rank zeta functions andSLn-zeta functions for curves

2020 ◽  
Vol 117 (12) ◽  
pp. 6398-6408
Author(s):  
Lin Weng ◽  
Don Zagier
Keyword(s):  
Zeta Function ◽  
Parabolic Subgroup ◽  
Vector Bundles ◽  
Reductive Group ◽  
Zeta Functions ◽  
Smooth Projective Curve ◽  
Maximal Parabolic Subgroup ◽  
Higher Rank ◽  
Semistable Vector Bundles

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case whenG=SLnand P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding ton=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

scholarly journals Computing motivic zeta functions on log smooth models

2019 ◽  
Vol 295 (1-2) ◽  
pp. 427-462 ◽  
Author(s):  
Emmanuel Bultot ◽  
Johannes Nicaise
Keyword(s):  
Zeta Function ◽  
Recent Work ◽  
Explicit Formula ◽  
Essential Role ◽  
Zeta Functions ◽  
Smooth Model ◽  
Motivic Zeta Functions ◽  
Motivic Zeta Function ◽  
Special Case

Abstract We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results.

scholarly journals Motivic zeta functions for curve singularities

2010 ◽  
Vol 198 ◽  
pp. 47-75 ◽  
Author(s):  
J. J. Moyano-Fernández ◽  
W. A. Zúňiga-Galindo
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Local Ring ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zeta Functions ◽  
Poincaré Series ◽  
Poincare Series ◽  
Motivic Zeta Functions ◽  
Curve Singularities

AbstractLet X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

scholarly journals Motivic zeta functions for curve singularities

2010 ◽  
Vol 198 ◽  
pp. 47-75
Author(s):  
J. J. Moyano-Fernández ◽  
W. A. Zúňiga-Galindo
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Local Ring ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zeta Functions ◽  
Poincaré Series ◽  
Poincare Series ◽  
Motivic Zeta Functions ◽  
Curve Singularities

AbstractLetXbe a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristicpbig enough. Given a local ringOp,x at a rational singular pointPofX, we attached a universal zeta function which is a rational function and admits a functional equation ifOp,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

scholarly journals Poles of maximal order of motivic zeta functions

2016 ◽  
Vol 165 (2) ◽  
pp. 217-243 ◽  
Author(s):  
Johannes Nicaise ◽  
Chenyang Xu
Keyword(s):  
Zeta Functions ◽  
Maximal Order ◽  
Motivic Zeta Functions

scholarly journals Motivic zeta functions of degenerating Calabi–Yau varieties

2017 ◽  
Vol 370 (3-4) ◽  
pp. 1277-1320 ◽  
Author(s):  
Lars Halvard Halle ◽  
Johannes Nicaise
Keyword(s):  
Zeta Functions ◽  
Motivic Zeta Functions

scholarly journals On the cohomology of certain subspaces of $$\text {Sym}^n(\mathbb {P}^1)$$ and Occam’s razor for Hodge structures

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Oishee Banerjee
Keyword(s):  
Zeta Functions ◽  
The Other ◽  
Grothendieck Ring ◽  
Hodge Structures ◽  
Symmetric Powers ◽  
Link Type ◽  
Occam’S Razor ◽  
Motivic Zeta Functions ◽  
Occam's Razor

AbstractVakil and Matchett-Wood (Discriminants in the Grothendieck ring of varieties, 2013. arXiv:1208.3166) made several conjectures on the topology of symmetric powers of geometrically irreducible varieties based on their computations on motivic zeta functions. Two of those conjectures are about subspaces of $$\text {Sym}^n(\mathbb {P}^1)$$ Sym n ( P 1 ) . In this note, we disprove one of them and prove a stronger form of the other, thereby obtaining (counter)examples to the principle of Occam’s razor for Hodge structures.

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