Asymptotic properties of zeta functions over finite fields

In this paper we study asymptotic properties of families of zeta and L-functions over finite fields. We do it in the context of three main problems: the basic inequality, the Brauer–Siegel type results and the results on distribution of zeroes. We generalize to this abstract setting the results of Tsfasman, Vlăduţ and Lachaud, who studied similar problems for curves and (in some cases) for varieties over finite fields. In the classical case of zeta functions of curves we extend a result of Ihara on the limit behaviour of the Euler–Kronecker constant. Our results also apply to L-functions of elliptic surfaces over finite fields, where we approach the Brauer–Siegel type conjectures recently made by Kunyavskii, Tsfasman and Hindry.     

Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation

We compute the homology of the Lie algebra W ₁ of (polynomial) vector fields on the line with coefficients in symmetric powers of its adjoint representation. We also list the results obtained so far for the homology with coefficients in tensor powers and, in turn, use them for partially computing the homology of the Lie algebra of W ₁-valued currents on the line.     

Separable polynomials over finite dimensional algebras

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality n^d or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly n^d distinct zeros in [Ktilde] ?_k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ?_k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with coefficients in A there is a non empty Zariski open subset consisting of separable polynomials; in other polynomials with coefficients in a finite dimensional algebra are “generically” separable.     

Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with involution. In this paper it is shown that over certain fields these invariants are sufficient to classify involutions up to conjugation. For algebras of low degree a classification is obtained over an arbitrary field. Received: 29 April 1999     

A natural graph of finite fields distinguishing between models

We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields. Certain basic notions in finite field theory have interpretations in terms of standard graph properties. We show that the graphs are connected and provide an estimate of their diameter. An accidental graph isomorphism is uncovered and proved. The smallest non-trivial Laplace eigenvalue is given some attention, in particular for a specific family of 8-regular graphs showing that it is not an expander. We introduce a regular covering graph and show that it is connected if and only if the root is primitive.     

Some elements of Lie-differential algebra and a uniform companion for large Lie-differential fields

In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin (see [E.R. Kolchin, Differential algebra and algebraic groups, in: Pure and Applied Mathematics, vol. 54, Academic Press, 1973]) by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 (2005) 163–190].

In particular it allows us to adapt the results of Tressl (see [M. Tressl, A uniform companion for large differential fields of characteristic zero, Trans. Amer. Math. Soc. 357 (10) (2005) 3933–3951]) by showing the existence of a theory of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie-differential fields extending a model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations.     

Infinite dimensional general linear groups have finite width

A group is said to have finite width whenever it has finite width with respect to each inverse-closed generating set. Bergman showed [1] that infinite symmetric groups have finite width and asked whether the automorphism groups of several classical structures have finite width, mentioning in particular infinite dimensional general linear groups over fields. In this article we prove that infinite dimensional general linear groups over arbitrary division rings have finite width. We consider the problem of finite width for other infinite dimensional classical groups.     

The structure of algebraic jordan algebras without nonzero nilpotent elements

An algebraic, linear Jordan algebra without nonzero nil-potent elements is proved to be a subdirect sum of prime Jordan algebras each of which has finite capacity or contains simple subalgebras of arbitrary capacity. If in addition the base field has nonzero character-istic or the algebra satisfies a polynomial identity, then each of the summands is determined to be simple of finite capacity. Further, it is proved that algebraic, PI Jordan algebras without nonzero nilpotent elements are locally finite in the sense that any finitely generated subalgebra has finite capacity.     

Model equation of the theory of solitons

We consider the hierarchy of integrable (1+2)-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain n arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable (1+1)-dimensional equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 29–45, October, 2007.     

Congruence preserving functions in the residue class rings of polynomials over finite fields

In this paper, as an analogue of the integer case, we define congruence preserving functions over the residue class rings of polynomials over finite fields. We establish a counting formula for such congruence preserving functions, determine a necessary and sufficient condition under which all congruence preserving functions are also polynomial functions, and characterize such functions.     

Exceptional crooked functions

Crooked functions are combinatorial objects of great interest. It is already known that the only monomial and binomial crooked functions are quadratic. In this paper, we investigate conditions on the shape of a polynomial to be crooked. Furthermore, the notion of exceptional crooked is introduced, similarly to those of APN or PN exceptional functions. Via a connection with algebraic varieties over finite fields, we provide non-existence results of exceptional crooked functions.     

On the Center of the Brauer Algebra

Using the determination of conjugacy classes in an earlier paper, we study the center of the Brauer algebra. In the case of finite groups, conjugacy class sums determine the center of the group algebra. In the case of the Brauer algebra the corresponding class sums only yield a basis of the centralizer of the symmetric group in the Brauer algebra. However, we exhibit an explicit algorithm to determine conditions for a centralizer element to be central and show how to compute a basis for the center using these methods. We will outline how this can be used to compute blocks over fields of arbitrary characteristic. We will also show that similar methods can be applied for computing a basis of the center of the walled Brauer algebra.     

Algebraic Aspects of the Dynamics of Quantum Multilevel Systems in the Projection Operator Technique

Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

关于有限域F_q上矩阵阶的研究

详细地研究了有限域 Fq上的矩阵的阶的问题 ,得到了相当理想的结果 .并给出一类矩阵方幂的极小多项式的求法     

有限域上与仿射多项式有关的一些多项式的可约性

在这篇文章中,研究了有限域上一些与仿射多项式有关的多项式的可约性.对于有限域Fp上不是xppt-x-1的仿射三项式,得到了这些三项式的一个明确的因式.完全确定了多项式g(xps-ax-b)在Fp[x]中的分解,这里g(x)是Fp[x]中一个不可约多项式.证明了Fp上次数相同的不可约多项式的全体可以构成一个正则图.同时给出了多项式g(xqs-x-b)在Fp[x]不可约因式的个数公式,这里g(x)是Fp上一个不可约多项式.     

An Embedding Theorem for Reduced Albert Algebras Over Arbitrary Fields

Extending two classical embedding theorems of Albert and Jacobson and Jacobson for Albert (exceptional simple Jordan) algebra over fields of characteristic not two to base fields of arbitrary characteristic, we show that any element of a reduced Albert algebra can be embedded into a reduced absolutely simple subalgebra of degree 3 and dimension 9 which may be chosen to be split if the Albert algebra was split to begin with.     

有限域上多项式集的简单分解

研究如何将任意有限域上的多项式集分解为有限多个简单列.为了解决这一问题,首先研究简单列和根理想之间的关系,然后基于已有的正则分解算法和有限域上理想的根的两种计算方法设计一个有限域上多项式集的简单分解算法.计算试验表明,文章给出的算法是有效的.     

Some multiplicative equations in finite fields

In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating the multiplicative energy of generalized arithmetic progressions in prime fields and of boxes in arbitrary finite fields. We obtain sharp bounds in more general scenarios than previously known. Our arguments extend some ideas of Konyagin and Bourgain and Chang into new settings.