The Alexander Polynomial of a Plane Curve Singularity and Integrals With Respect to the Euler Characteristic

Campillo, A.; Delgado, F.; Guseĭn-Zade, S. M.

doi:10.1142/s0129167x03001703

Cited by 36 publications

(66 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…As a consequence of the results of Campillo, Delgado, and GuseinZade (see [5], [6], [7]) and Lemma 4, we have χ(Z(T, O)) = ς f (T ), the zeta function of the monodromy ς f (T ) associated to the germ of function f : (C 2 , 0) → (C, 0). By Remark 9 and Lemma 7, we have…”

Section: And For Any S = S(o) We Havementioning

confidence: 76%

Motivic zeta functions for curve singularities

Moyano‐Fernández¹,

Zúñiga–Galindo²

2010

Nagoya Mathematical Journal

View full text Add to dashboard Cite

Abstract. Let 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. §1. Introduction Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a finite field F q . In [29], the second author introduced a zeta function Z(Ca(X), T ) associated to the effective Cartier divisors on X. Other types of zeta functions associated to singular curves over finite fields were introduced in [15], [16], [23], [24], and [30]. The zeta function Z(Ca(X), T ) admits an Euler product with nontrivial factors at the singular points of X. If P is a rational singular point of X, then the local factor Z Ca(X) (T, q, O P,X ) at P is a rational function of T depending on q and the completion O P,X of the local ring O P,X of X at P . If the residue field of O P,X is not too small, then Z Ca(X) (T, q, O P,X ) depends only on the semigroup of O P,X (see [29, Theorem 5.5]). Thus, if O P,X ∼ = F q Jx, yK/(f (x, y)),

show abstract

Section: And For Any S = S(o) We Havementioning

confidence: 76%

Motivic zeta functions for curve singularities

Moyano‐Fernández¹,

Zúñiga–Galindo²

2010

Nagoya Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…The proof essentially repeats the arguments from [3], [9]. One uses the representation of the Poincaré series as an integral with respect to the Euler characteristic.…”

Section: Definitionmentioning

confidence: 89%

Alexander polynomials and Poincaré series of sets of ideals

Guseĭn-Zade

Delgado

Campillo

2011

Funct Anal Its Appl

Self Cite

View full text Add to dashboard Cite

“…Доказательство по существу повторяет рас-суждения из [3], [8]. Используется представление ряда Пуанкаре в виде инте-грала по отношению к эйлеровой характеристике.…”

Section: теорема 1 имеет место равенствоunclassified

“…Следующее понятие инте-грала по отношению к эйлеровой характеристике по пространству PO V,0 было описано, например, в [3].…”

unclassified