Recursively enumerable sets of polynomials over a finite field

We prove that a relation over is recursively enumerable if and only if it is Diophantine over . We do this by first constructing a model of in , where n is represented by Zⁿ. In a second step, we show that it suffices to eliminate a bounded universal quantifier. Then finally, the hardest part of the proof is to show that we can eliminate this quantifier.     

有限域上单变量代数函数域中的素数定理

设q是素数的幂次，Fq为一有限域；F为Fq上的单变量代数函数域．在这篇文章中我们证明了下面的素数定理，πF（x）=1/（q-1）.x/logqx＋O（x/log^2qx）.x=q^n→∞其中logqx以q为底的对数，这一结果改进了M．Kruse，H．Stichtenoth的结果．     

Discrete Logarithm Based Cryptosystems in Quadratic Function Fields of Characteristic 2

We present a key exchange scheme similar to that of Diffie and Hellman using the infrastructure of quadratic function fields of even characteristic. This is a modification of the results of Scheidler, Stein and Williams who used quadratic function fields of odd characteristic. We also extend these results to give a digital signature scheme similar to that of ElGamal. These schemes are possible in this structure even though it is not a group. Finally we examine the security of such systems, and give a possible attack based on Pohlig and Hellman's attack on discrete logarithms in finite groups.     

保特征2的域上幂等矩阵的线性算子

设Ｆ是特征２的域．本文确定Ｆ上ｎ×ｎ矩阵保幂等性的可逆的线性算子的半群Ｈ（Ｆ）的结构，同时回答了文献［１］提出的两个未解决的问题．     

On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields . The Weil estimate of such sums is well known: , where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, , for which attains the largest possible value and, in particular, .     

On Polynomials of Special Form over a Finite Field of Odd Characteristic Attaining the Weil Bound

An explicit construction of polynomials over a finite field of odd characteristic, for which the absolute value of trigonometric sums attains the Weil bound, is based on the construction of cyclic matrices of given rank. Dickson polynomials of the second kind play an essential role in the study of such matrices.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 16–25.Original Russian Text Copyright © 2005 by L. A. Bassalygo, V. A. Zinov’ev.     

On Recognition by Spectrum of Finite Simple Linear Groups over Fields of Characteristic 2

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L _n (2^k) are recognizable by spectrum for n = 2^m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005.     

Uniform Families of Polynomial Equations Over a Finite Field and Structures Admitting an Euler Characteristic of Definable Sets

Over a fixed finite field F_p, families of polynomial equations for i = 1,..., k_N, that areuniformly determined by a parameter N, are considered. The notionof a uniform family is defined in terms of first-order logic.A notion of an abstract Euler characteristic is used to givesense to a statement that the system has a solution for infiniteN, and a statement linking the solvability of a linear systemfor infinite N with its solvability for finite N is proved.This characterisation is used to formulate a criterion yieldingdegree lower bounds for various ideal membership proof systems(for example, Nullstellensatz and the polynomial calculus).Further, several results about Euler structures (structureswith an abstract Euler characteristic) are proved, and the caseof fields, in particular, is investigated more closely. 1991Mathematics Subject Classification: primary 03F20, 12L12, 15A06;secondary 03C99, 12E12, 68Q15, 13L05.     

A Review on the Isomorphism Classes of Hyperelliptic Curves of Genus 2 over Finite Fields Admitting a Weierstrass Point

The reduced equations for the isomorphism classes of hyperelliptic curves of genus 2 admitting a Weierstrass point over a finite field of arbitrary characteristic, are shown and the number of such classes is included. This work picks up in a unified way a series of previous results published by several authors by using different methodologies. These classifications are of interest in designing and implementing of hyperelliptic curve cryptosystems.     

利用有限域上伪辛几何中一类2维非迷向子空间构作PBIB设计

设Ｆｑ是特征为２的有限域，本文利用Ｆｑ上２ｖ＋２维伪辛几何中包含固定的１维非迷向子空间的一类的２维非迷向子空间作处理，构作了具有２（ｑ－１）个结合类的结合方法和ＰＢＩＢ设计，并计算了相应的参数。     

奇特征有限正交空间中非迷向子空间的Critical问题

利用奇特征正交空间的性质及计数定理在奇特征正交空间中研究了非迷向子空间的Critical问题,得到了相应的计数公式和Critical指数.     

保特征2域上上三角矩阵群逆的线性映射

设F是一个特征2且至少含有5个元素的域,n≥2是一个正整数．令Mn（F）和Tn（F）分别F上的全矩阵空间和上三角矩阵空间．我们首先刻划从Tn（F）到Mn（F）的保矩阵群逆的所有线性单射，由此从Tn（F）到自身的所有保矩阵群逆的线性双射被刻划．     

Classification of Finite Factor Representations of the (2m+1)-Dimensional Heisenberg Group over a Countable Field of Finite Characteristic

For a field F that is the direct limit of an increasing chain of finite fields, we describe the Bratteli diagram, the finite complex factor representations, the Plancherel formula, and the projective modules of the corresponding Heisenberg group.     

On Recognition of the Finite Simple Orthogonal Groups of Dimension 2m, 2m+1, and 2m+2 over a Field of Characteristic 2

The spectrum (G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if HG for every finite group H such that (H)=(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum.     

Isotropy of 5-dimensional quadratic forms over the function field of a quadric in characteristic 2

We complete a classification of quadratic forms over a field of characteristic 2 of type (1,3) that become isotropic over the function field of a quadric.