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．     

On the Number of Points of Elliptic Curves over a Finite Field and a Problem of B. Segre

Several theorems are studied concerning the number of points of an elliptic curve with a Legendre form on a finite field, in order to analyse the distribution of regular and pseudoregular points in relation to a hyperbola in a finite affine plane.     

Subtleties in the Distribution of the Numbers of Points on Elliptic Curves Over a Finite Prime Field

Three questions concerning the distribution of the numbers ofpoints on elliptic curves over a finite prime field are considered.First, the previously published bounds for the distributionare tightened slightly. Within these bounds, there are wildfluctuations in the distribution, and some heuristics are discussed(supported by numerical evidence) which suggest that numbersof points with no large prime divisors are unusually prevalent.Finally, allowing the prime field to vary while fixing the fieldof fractions of the endomorphism ring of the curve, the orderof magnitude of the average order of the number of divisorsof the number of points is determined, subject to assumptionsabout primes in quadratic progressions. There are implications for factoring integers by Lenstra's ellipticcurve method. The heuristics suggest that (i) the subtletiesin the distribution actually favour the elliptic curve method,and (ii) this gain is transient, dying away as the factors tobe found tend to infinity.     

Prime Divisors of the Number of Rational Points on Elliptic Curves with Complex Multiplication

Let E/Q be an elliptic curve. For a prime p of good reduction,let E(F_p) be the set of rational points defined over the finitefield F_p. Denote by (#E(F_p)) the number of distinct prime divisorsof #E(F_p). For an elliptic curve with complex multiplication,the normal order of (#E(F_p)) is shown to be log log p. The normalorder of the number of distinct prime factors of the exponentof E(F_p) is also studied. 2000 Mathematics Subject Classification11N37, 11G20.     

The Number of Nonisomorphic Two-dimensional Algebras over a Finite Field

In this paper, we derive explicit formulas for the number of nonisomorphic two-dimensional nonassociative algebras, possibly without a unit, over a finite field. The proof combines the first author’s general classification theory of two-dimensional nonassociative algebras over arbitrary base fields with elementary counting arguments which are primarily addressed to the problem of determining the number of orbits of a finite set acted upon by the group of integers mod 2. The number of nonisomorphic two-dimensional division algebras will also be determined.     

The Number of Prime Factors of the Scale Function on a Compactly Generated Group is Finite

It is shown that for each compactly generated totally disconnectedlocally compact group G, there is a finite number of prime numbers,p₁, p₂, ..., p_n, such that the scale function s : G N satisfies, where x G.     

合数是绝对假素数的充要条件

利用正整数模的特征数这一新概念给出了合数是绝对假素数的充要条件。以此为据，证明了绝对假素数是奇数，它无异于1的平方因数，并且至少是三个互异的奇素数的乘积；还给出了两个绝对假素数或两个大于1的奇数的乘积是绝对假素数的充要条件。     

On the Number of Rational Points of Hypersurfaces over Finite Fields

We prove the existence of hypersurfaces defined over finite fields having a prescribed number of -rational points and a prescribed number of non-singular points. Moreover, some results on -rational intersections between plane curves, lines and conics, are given. Received: June 1, 2006. Revised: August 1, 2007.     

More Points Than Expected on Curves over Finite Field Extensions

On average, there are q^r+o(q^r/2) F_q^r-rational points on curves of genus g defined over F_q^r. This is also true if we restrict our average to genus g curves defined over F_q, provided r is odd or r>2g. However, if r=2,4,6,… or 2g then the average is q^r+q^r/2+o(q^r/2). We give a number of proofs of the existence of these q^r/2 extra points, and in some cases give a precise formula, but we are unable to provide a satisfactory explanation for this phenomenon.     

The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.with an Appendix by Jean-Pierre Serre     

On the Number of Solutions of an Equation Over a Finite Field

The Stöhr–Voloch approach is used to obtain a newbound for the number of solutions in (F_q)² of an equation f(X,Y) = 0, where f(X, Y) is an absolutely irreducible polynomialwith coefficients in a finite field F_q.     

On the Number of Cyclic Subgroups of Prime Order in the Group of Diagonal Matrices over a Circular Field

Mathematical Notes -     

The Witt Ring of a Smooth Projective Curve Over a Finite Field

In this paper, we calculate the Witt ring W(C) of a smooth geometrically connected projective curve C over a finite field with characteristic other than 2. We view W(C) as a subring of W(k(C)) where k(C) is the function field of C. The calculation is then completed using classical results for bilinear spaces over fields.     

数域上的线性无关性

本文给出一组复数在数域上线性无关的一个判别法则及它对于超越数论的一个应用．     

On the Number of Galois Points for a Plane Curve in Positive Characteristic

We study Galois points for a plane smooth curve C ? P ² of degree d ≥ 4 in characteristic p > 2. We generalize Yoshihara's result on the number of inner (resp., outer) Galois points to positive characteristic under the assumption that d ? 1 (resp., d ? 0) modulo p. As an application, we also find the number of Galois points in the case that d = p.     

非线性不可约特征标维数相等且为素数个的有限群

对于具有素数个非线性不可约特征标且它们的维数相等的有限群,我们给出一个分类.     

The Number of Permutation Polynomials of a Given Degree Over a Finite Field

We relate the number of permutation polynomials in F_q[x] of degree d≤q−2 to the solutions (x₁,x₂,…,x_q) of a system of linear equations over F_q, with the added restriction that x_i≠0 and x_i≠x_j whenever i≠j. Using this we find an expression for the number of permutation polynomials of degree p−2 in F_p[x] in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity. This leads to nontrivial bounds for the number of such permutation polynomials. We provide numerical examples to illustrate our method and indicate how our results can be generalised to polynomials of other degrees.     

On the Exceptional Set of the Sum of a Prime Number and a Fixed Degree of a Prime Number

Let X be a sufficiently great real number and M denote the set of natural numbers not exceeding X which cannot be written as a sum of a prime and a fixed degree of a prime number from the arithmetical progression with difference d. Let Ed(X) = cardM. We obtain a new numerical degree estimate for the set Ed(X) and an estimate from below for the number of presentations of n ∉ M in the specified type. The proven estimates refine the generalization for an arithmetical progression of results earlier got by V.A. Plaksin.