scholarly journals POINCARÉ SERIES OF NON-DIVISORIAL VALUATIONS ON TWO-DIMENSIONAL FUNCTION FIELDS

2015 ◽  
Vol 36 (3) ◽  
pp. 269-277
Author(s):  
Charles Li ◽  
Hans Schoutens
Keyword(s):  
Function Fields ◽  
Poincaré Series ◽  
Two Dimensional ◽  
Poincare Series ◽  
Divisorial Valuations

scholarly journals POINCARÉ SERIES FOR SEVERAL PLANE DIVISORIAL VALUATIONS

2003 ◽  
Vol 46 (2) ◽  
pp. 501-509 ◽  
Author(s):  
F. Delgado ◽  
S. M. Gusein-Zade
Keyword(s):  
Plane Curve ◽  
Poincaré Series ◽  
Finite Collection ◽  
Curve Singularity ◽  
Motivic Integration ◽  
Poincare Series ◽  
Plane Curve Singularity ◽  
Functions Of Two Variables ◽  
Irreducible Components ◽  
Divisorial Valuations

AbstractWe compute the (generalized) Poincaré series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincaré series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.AMS 2000 Mathematics subject classification: Primary 14B05; 16W70

Theta Series, Eisenstein Series and Poincaré Series over Function Fields

2004 ◽  
Vol 56 (2) ◽  
pp. 406-430 ◽  
Author(s):  
Ambrus Pál
Keyword(s):  
Eisenstein Series ◽  
Finite Fields ◽  
Function Fields ◽  
Theta Series ◽  
Poincaré Series ◽  
Poincare Series ◽  
Basic Properties

AbstractWe construct analogues of theta series, Eisenstein series and Poincaré series for function fields of one variable over finite fields, and prove their basic properties.

scholarly journals POINCARÉ SERIES OF COLLECTIONS OF PLANE VALUATIONS

2010 ◽  
Vol 21 (11) ◽  
pp. 1461-1473 ◽  
Author(s):  
A. CAMPILLO ◽  
F. DELGADO ◽  
S. M. GUSEIN-ZADE ◽  
F. HERNANDO
Keyword(s):  
Plane Curve ◽  
Poincaré Series ◽  
Poincare Series ◽  
Multi Index ◽  
Plane Curve Singularities ◽  
Functions Of Two Variables ◽  
Curve Singularities ◽  
Divisorial Valuations ◽  
General Collection ◽  
Reducible Plane

In earlier papers there were given formulae for the Poincaré series of multi-index filtrations on the ring [Formula: see text] of germs of functions of two variables defined by collections of valuations corresponding to (reducible) plane curve singularities and by collections of divisorial ones. It was shown that the Poincaré series of a collection of divisorial valuations determines the topology of the collection of divisors. Here we give a formula for the Poincaré series of a general collection of valuations on the ring [Formula: see text] centered at the origin and prove a generalization of the statement that the Poincaré series determines the topology of the collection.

scholarly journals Poincaré series of multiplier ideals in two-dimensional local rings with rational singularities

2017 ◽  
Vol 304 ◽  
pp. 769-792 ◽  
Author(s):  
Maria Alberich-Carramiñana ◽  
Josep Àlvarez Montaner ◽  
Ferran Dachs-Cadefau ◽  
Víctor González-Alonso
Keyword(s):  
Poincaré Series ◽  
Local Rings ◽  
Two Dimensional ◽  
Poincare Series ◽  
Multiplier Ideals ◽  
Rational Singularities
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