S. M. Gusein-Zade, I. Luengo, A. Melle-Hernandez
We use the notion of Milnor fibres of the germ of a meromorphic function and the method of partial resolutions for a study of topology of a polynomial map at infinity (mainly for calculation of the zeta-function of a monodromy). It gives effective methods of computation of the zeta-function for a number of cases and a criterium for a value to be atypical at infinity.
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By using sheaf-theoretical methods such as constructible sheaves, we generalize the formula of Libgober-Sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. In particular, some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained.
We explicitly construct a universal A-infinity deformation of Batalin-Vilkovisky algebras, with all coefficients expressed as rational sums of multiple zeta values. If the Batalin-Vilkovisky algebra that we start with is cyclic, then so is the A-infinity deformation. Moreover, the adjoint action of the odd Poisson bracket acts by derivations of the A-infinity structure. The construction conjecturally defines a new presentation of the Grothendieck-Teichmueller Lie algebra.
In a recent paper Zúñiga-Galindo and the author begun the study of the local zeta functions for Laurent polynomials. In this work we continue this study by giving a very explicit formula for the local zeta function associated to a Laurent polynomial $f$ over a $p-$adic field, when $f$ is weakly non-degenerate with respect to the Newton polytope of $f$ at infinity.
In earlier work, the authors described a relation between the Poincaré series and the classical monodromy zeta function corresponding to a quasihomogeneous polynomial. Here we formulate an equivariant version of this relation in terms of the Burnside rings of finite abelian groups and their analogues.
In this paper, we prove the rationality of Igusa's local zeta functions of semiquasihomogeneous polynomials with coefficients in a non-archimedean local field K. The proof of this result is based on Igusa's stationary phase formula and some ideas on Neron p-desingularization.
We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, which are Mukai dual for zeta=0 and infinity, amd mirror partners for zeta not equal to 0 and infinity. Moreover, the Fourier-Mukai equivalence D^b(M,cal{I}_0) -> D^b(M,cal{J}_0) induces an isomorphism phi_T between the spaces of first order deformations of (M,cal{I}_0) and (M,cal{J}_0) as generalized complex manifolds, and the deformations (M…
An invertible polynomial in $n$ variables is a quasihomogeneous polynomial consisting of $n$ monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau--Ginzburg models, P.~Berg\-lund, T.~Hübsch and M.~Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and an abelian group $G$ of its symmetries together with a dual pair $(\widetilde{f}, \widetilde{G})$. Here we study the reduced orbifold zeta functions of dual pairs $(f,G)$ and $(\widetilde{f}, \widetilde{G})$ and sh…
In this article, we study local zeta functions attached to Laurent polynomials over p-adic fields, which are non-degenerate with respect to their Newton polytopes at infinity. As an application we obtain asymptotic expansions for p-adic oscillatory integrals attached to Laurent polynomials. We show the existence of two different asymptotic expansions for p-adic oscillatory integrals, one when the absolute value of the parameter approaches infinity, the other when the absolute value of the parameter approaches zero. These two asymptotic expansions are controlled by the poles of twisted local ze…
On the coefficients of the Zeta-function's L-polynomial for algebraic function fields over finite constant fields
本文研究了代数函数域上有限常数域的ζ函数L多项式系数的显式公式，并推导了类数表达式，同时给出了缺陷为2的曲线应用。
Deformations of polynomials and their zeta functions
本文研究了多项式解析族的单值变换的zeta函数，将其表示为欧拉示性数的积分，并展示了计算实例。
A generalization of Arnold's strange duality to invertible polynomials in three variables by the first author and A.Takahashi includes the following relation. For some invertible polynomials $f$ the Saito dual of the reduced monodromy zeta function of $f$ coincides with a formal "root" of the reduced monodromy zeta function of its Berglund-Hübsch transpose $f^T$. Here we give a geometric interpretation of "roots" of the monodromy zeta function and generalize the above relation to all non-degenerate invertible polynomials in three variables and to some polynomials in an arbitrary number of vari…
In general, the L-polynomial of a curve of genus $g$ is determined by $g$ coefficients. We show that the L-polynomial of a supersingular curve of genus $g$ is determined by fewer than $g$ coefficients.
The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez--Villegas transform ([Ro--V]) a polynomial satisfying the functional equation of zeta type and having nontrivial zeros only on the middle line of its critical strip. The second part discusses Chebyshev lambda--structure of the polynomial ring as Borger's descent data to $\bold{F}_1$ and suggests its role in possible relation of $Γ_{\bold{R}}$--factor to "real geometry over $\bold{F}_1$" (cf. also [CoCons2]).
On Motivic Zeta Functions and Stringy E-function via Embedded $\mathbb{Q}$-Resolution
该研究通过嵌入$\u211a$-消解方法，推导半拟齐次奇点的动机Zeta函数公式及二维极点，提出弦E函数的新公式并用于半拟齐次、非退化多项式的弦E函数计算。
For a generic (polynomial) one-parameter deformation of a complete intersection, there is defined its monodromy zeta-function. We provide explicit formulae for this zeta-function in terms of the corresponding Newton polyhedra in the case the deformation is non-degenerate with respect to its Newton polyhedra. Using this result we obtain the formula for the monodromy zeta-function at the origin of a polynomial on a complete intersection, which is an analog of the Libgober--Sperber theorem.
We present expressions for the Weierstrass zeta-function and related elliptic functions by rapidly converging series. These series arise as triple products in the A-infinity category of an elliptic curve.
Poles of real motivic zeta functions for curves
本文研究了实曲线的实motivic zeta函数的极点，通过分析实拓扑zeta函数的极点特性，揭示其作为吹胀-Nash等价不变量的数学性质。
In 2005, Kayal suggested that Schoof's algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pila's algorithm for abelian varieties.
To 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 $q^{-s}$…
The paper gives a full description of the growth of the gradient of a polynomial in two complex variables at infinity near any fiber of the polynomial
We address questions posed by Lorenzini about relations between Jacobians, Tutte polynomials, and the Brill-Noether theory of finite graphs, as encoded in his two-variable zeta functions. In particular, we give examples showing that none of these invariants is determined by the other two.
For p prime, we give an explicit formula for Igusa's local zeta function associated to a polynomial mapping f=(f_1,...,f_t): Q_p^n -> Q_p^t, with f_1,...,f_t in Z_p[x_1,...,x_n], and an integration measure on Z_p^n of the form |g(x)||dx|, with g another polynomial in Z_p[x_1,...,x_n]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F_p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results of Denef - Hoornaer…
We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Zúñiga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for…
For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalization of a conjecture of Igusa - Denef - Loeser predicts that every pole of its topological zeta function is a root of its Bernstein-Sato polynomial. However, typically only a few roots are obtained this way. Following ideas of Veys, we study the following question. Is it possible to find a collection G of polynomials g in C[x_1,...,x_n], such that, for all g in G, every pole of the topological zeta function associated to I and the volume form gdx on the affine n-space, is a root of the Bernstein-Sato polynomial of I, and su…
Riemann Hypothesis for Non-Abelian Zeta Functions of Genus 2 Curves
本文针对有限域上亏格2曲线的Weng非阿贝尔zeta函数，利用半稳定丛模空间几何性质、高秩结果与渐近分析，证明了当秩n趋于无穷时黎曼猜想成立。
A polynomial counterpart of the Seiberg-Witten invariant associated with a negative definite plumbed 3-manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta-function defined by the combinatorics of the manifold. In this article we give an algorithm, based on multivariable Euclidean division of the zeta-function, for the explicit calculation of the polynomial, in particular for the Seiberg--Witten invariant.
The aim of this paper is to describe explicitly the poles of the meromorphic continuation of the Igusa local zeta function associated to several polynomials. Using resolution of singularities is possible to express the Igusa's local zeta function as a finite sum of p-adic monomial integrals. We compute these monomial integrals using techniques of toroidal geometry. In this way, we obtain an explicit list for the candidates to poles of the local zeta function associated to several polynomials.
We will introduce a regularization for $p$-adic multiple zeta values and show that the generalized double shuffle relations hold. This settles a question raised by Deligne, given as a project in Arizona winter school 2002. Our approach is to use the theory of Coleman functions on the moduli space of genus zero curves with marked points and its compactification. The main ingredients are the analytic continuation of Coleman functions to the normal bundle of divisors at infinity and definition of a special tangential base point on the moduli space.
Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacob…
In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary k poles of prescribed orders, such that for every geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of a p-cover of the projective line totally ramified at arbitrary k points of prescribed orders has an asymptotic gener…
We study the asymptotics of fundamental solutions of p-adic pseudo-differential equations connected with homogeneous polynomials by using techniques of local zeta functions theory.
Canonicalizing zeta generators: genus zero and genus one
本文针对零亏格与一亏格的ζ生成元开展规范化研究，通过多重ζ值对偶空间确定规范多项式，构造对应同构，解决一亏格生成元非几何部分的定义歧义，建立了两类亏格下ζ生成元的关联，连通了两类数学对象的构造。
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