Cohomological Interpretation of the Motivic Zeta Function

2016 ◽  
pp. 129-140
Author(s):  
Lars Halvard Halle ◽  
Johannes Nicaise
Keyword(s):  
Zeta Function ◽  
Cohomological Interpretation ◽  
Motivic Zeta Function

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 Zeta Functions and ‘Kontsevich Invariants’ on Singular Varieties

2001 ◽  
Vol 53 (4) ◽  
pp. 834-865 ◽  
Author(s):  
Willem Veys
Keyword(s):  
Zeta Function ◽  
Regular Function ◽  
Zeta Functions ◽  
Singular Locus ◽  
Normal Surface ◽  
Grothendieck Ring ◽  
Singular Varieties ◽  
Hodge Numbers ◽  
Topological Zeta Function ◽  
Motivic Zeta Function

AbstractLet X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any ℚ-Gorenstein variety X we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective ℚ-Cartier divisors on X whose support contains the singular locus of X.

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 The Newton tree: geometric interpretation and applications to the motivic zeta function and the log canonical threshold

2015 ◽  
Vol 159 (3) ◽  
pp. 481-515 ◽  
Author(s):  
PIERRETTE CASSOU-NOGUÈS ◽  
WILLEM VEYS
Keyword(s):  
Zeta Function ◽  
Newton Polygon ◽  
Geometric Interpretation ◽  
Canonical Model ◽  
Newton Polygons ◽  
Log Canonical Threshold ◽  
Newton Algorithm ◽  
Log Canonical ◽  
Motivic Zeta Function ◽  
The Ideal

AbstractLet${\mathcal I}$be an arbitrary ideal in${\mathbb C}$[[x,y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal Newton tree to${\mathcal I}$, related to using good coordinates in the Newton algorithm, and show that it has a conceptual geometric interpretation in terms of the log canonical model of${\mathcal I}$. We also compute the log canonical threshold from a Newton polygon and strengthen Corti's inequalities.

scholarly journals The motivic Igusa zeta function of a space monomial curve with a plane semigroup

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hussein Mourtada ◽  
Willem Veys ◽  
Lena Vos
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Closed Formula ◽  
Generic Fiber ◽  
Jet Schemes ◽  
Irreducible Components ◽  
The Family ◽  
Generic Curve ◽  
Motivic Zeta Function ◽  
Monomial Curve

Abstract In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to the number of poles of the motivic zeta function associated with a generic curve in the family.

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 The motivic zeta function and its smallest poles

2007 ◽  
Vol 317 (2) ◽  
pp. 851-866
Author(s):  
Dirk Segers ◽  
Lise Van Proeyen ◽  
Willem Veys
Keyword(s):  
Zeta Function ◽  
Motivic Zeta Function

Motivic zeta functions in additive monoidal categories

2011 ◽  
Vol 9 (3) ◽  
pp. 459-473 ◽  
Author(s):  
Kenichiro Kimura ◽  
Shun-ichi Kimura ◽  
Nobuyoshi Takahashi
Keyword(s):  
Zeta Function ◽  
Monoidal Category ◽  
Zeta Functions ◽  
Monoidal Categories ◽  
Symmetric Monoidal Category ◽  
Motivic Zeta Functions ◽  
Motivic Zeta Function ◽  
Coefficient Ring

AbstractLet C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C. We study the problem of rationality of the motivic zeta function ζx(t) of X. Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζx(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.

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