On the period length of the generalized Lagrange algorithm

The generalized Lagrange algorithm is a number geometric generalization of Lagrange's continued fraction method for computing fundamental unit and class number of real quadratic number fields. This algorithm yields a system of fundamental units and the class number of an arbitrary algebraic number field by means of computing cycles of reduced ideals. In this paper we prove that the cardinality of a cycle of reduced ideals in an ideal class of an order of an algebraic number field is O(R), where R is the regulator of this order, and where the O-constant only depends on the degree of the field. We also give a lower bound on this cardinality.     

A generalization of Berwick's unit algorithm

In 1913 W. E. H. Berwick published an algorithm for finding the fundamental unit of a cubic field with negative discriminant. His method relied heavily on the geometry of such fields and was less efficient than the well-known algorithm of Voronoi. In the present paper we show that the use of cubic geometry is not necessary and also that Berwick's method can be generalized. We present a periodic algorithm for finding a maximal set of independent units in an arbitrary algebraic number field.     

The class-number one problem for some real cubic number fields with negative discriminants

We prove that there are effectively only finitely many real cubic number fields of a given class number with negative discriminants and ring of algebraic integers generated by an algebraic unit. As an example, we then determine all these cubic number fields of class number one. There are 42 of them. As a byproduct of our approach, we obtain a new proof of Nagell's result according to which a real cubic unit ?>1 of negative discriminant is generally the fundamental unit of the cubic order Z[?].     

A generalization of Voronoi's unit algorithm II

It is proved that the generalized Voronoi algorithm developed in Part I of this paper computes the fundamental units of all fields of unit rank 2, i.e., of the totally real cubic fields, of the quartic fields with two real conjugate fields, of the quintic fields with only one real conjugate field, and of the totally complex sextic fields.     

连续代数扩域上多项式因式分解的Trager算法

多项式的因式分解是符号计算中最基本的算法,二十世纪六十年代开始出现的关于多项式因式分解的工作被认为是符号计算领域的起源．目前多项式的因式分解已经成熟,并已在Maple等符号计算软件中实现,但代数扩域上的因式分解算法还有待进一步改进．代数扩域上的基本算法是Trager算法．Weinberger等提出了基于Hensel提升的算法．这些算法是在单个扩域上做因式分解．而在吴零点分解定理中,多个代数扩域上的因式分解是非常基本的一步,主要用于不可约升列的计算．为了解决这一问题,吴文俊,胡森、王东明分别提出了基于方程求解的多个扩域上的因式分解算法．王东明、林东岱提出了另外一个算法Trager算法相似,将问题化为有理数域上的分解．他们应用了吴的三角化算法,因此算法的终止性依赖于吴方法的计算．支丽红则将提升技巧用于多个扩域上的因式分解算法．本文将Trager的算法直接推广为连续扩域上的因式分解,只涉及结式计算与有理数域上的因式分解,给出了多个代数扩域上的因式分解一个直接的算法．     

The class number one problem for some totally complex quartic number fields

We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit εu generally generates the unit group of the quartic order Z[εu].     

Extension of Hilbert's tenth problem to some algebraic number fields

We extend the solution of Hilbert's tenth problem to algebraic number fields having one pair of complex conjugated embeddings. The proof is based on the extended method of J. Denef used for totally real algebraic number fields.     

Elements with prime and small indices in bicyclic biquadratic number fields

We give necessary and sufficient conditions for the existence of primitive algebraic integers with index A in totally complex bicyclic biquadratic number fields where A is an odd prime or a positive rational integer at most 10. We also determine all these elements and prove that there are infinitely many totally complex bicyclic biquadratic number fields containing elements with index A.     

Computational Complexity in Algebraic Systems

Computability and computational complexity were first considered over the fields of real and complex numbers and generalized to arbitrary algebraic systems. This article approaches the theory of computational complexity over an arbitrary algebraic system by taking computability over the list extension for a computational model of it. We study the resultant polynomial complexity classes and mention some NP-complete problems.     

y^2=x^3＋27x-62上的整数点

使用代数数论和p-adic分析,我们找到了椭圆曲线y^2=x^3＋27x-62上所有的整数点．我们给出了一个全虚四次域的子环上计算基本单位和二次代数数“不相关分解”的方法．     

Diophantine equations and class numbers

The goals of this paper are to provide: (1) sufficient conditions, based on the solvability of certain diophantine equations, for the non-triviality of the class numbers of certain real quadratic fields; (2) sufficient conditions for the divisibility of the class numbers of certain imaginary quadratic fields by a given integer; and (3) necessary and sufficient conditions for an algebraic integer (which is not a unit) to be the norm of an algebraic integer in a given extension of number fields.     

Representation of algebraic integers by ternary quadratic forms

The discrete ergodic method is generalized to totally positive-ternary quadratic forms over totally real algebraic number fields. We obtain estimates for the number of representations of elements in maximal orders of such number fields which are precise in the sense of the order of growth. We prove that the representations are asymptotically uniformly distributed with respect to a given module.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 157–168, 1983.     

On the theorem of Barban and Davenport-Halberstam in algebraic number fields

The Barban-Davenport-Halberstam theorem is generalized to arbitrary algebraic number fields of finite degree over the rationals. As an application, we consider a divisor problem and a result of Elliott and Halberstam concerning the least prime in an arithmetic progression.     

The number of steps in the euclidean algorithm over complex quadratic fields

We obtain upper and lower bounds for the number of divisions in the Euclidean algorithm, for almost all pairs of algebraic integers lying in the complex quadratic fields (–m), form=1, 2, 3, 7 and 11. In addition, the order of the average length for almost all such pairs is deduced.     

Non-galois cubic fields which are euclidean but not norm-euclidean

Weinberger in 1973 has shown that under the Generalized Riemann Hypothesis for Dedekind zeta functions, an algebraic number field with infinite unit group is Euclidean if and only if it is a principal ideal domain. Using a method recently introduced by us, we give two examples of cubic fields which are Euclidean but not norm--Euclidean.

     

Dynamic interactive stabilization of the switching Kumar-Seidman system

A dynamic control protocol for the Kumar-Seidman flexible production system represented in general algebraic form is suggested. The ultimate purpose of the study is to minimize the total amount of work per unit time. The suggested protocol is proved to generate the required periodic process as a global attractor. In order to substantiate convergence, a number of statements of classical Frobenius-Perron theory are generalized to monotone piecewise affine nonlinear operators. A new method for exciting the required production cycles in the spirit of classical Poincaré’s method is suggested. The approach is based on a new stability criterion for an equilibrium of a discrete stationary system. A dynamic control protocol for the Kumar-Seidman flexible production system represented in general algebraic form is suggested and proved to generate the required periodic process as a global attractor. In order to substantiate convergence, a number of statements of classical Frobenius-Perron theory are generalized to monotone piecewise affine nonlinear operators.     

Littlewood Pisot numbers

A Pisot number is a real algebraic integer, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is a Littlewood polynomial, one with {+1,-1}-coefficients, and shows that they form an increasing sequence with limit 2. It is known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Littlewood polynomials. Finally, we prove that every reciprocal Littlewood polynomial of odd degree n?3 has at least three unimodular roots.     

Diophantine equations over global function fields IV: S-unit equations in several variables with an application to norm form equations

We describe an efficient algorithm to calculate all solutions of unit equations in several variables over global function fields. Note that using the present tools it is not possible to solve completely unit equations in more than two variables over number fields. In the function field case such equations are completely solved here for the first time. As a typical application we determine all solutions of norm form equations.     

Arithmetic Properties of Generalized Hypergeometric <Emphasis Type="Italic">F</Emphasis>-Series

A generalization of the Siegel–Shidlovskii method in the theory of transcendental numbers is used to prove the infinite algebraic independence of elements (generated by generalized hypergeometric series) of direct products of fields \(\mathbb{K}_v\), which are completions of an algebraic number field \(\mathbb{K}\) of finite degree over the field of rational numbers with respect to valuations v of \(\mathbb{K}\) extending p-adic valuations of the field ? over all primes p, except for a finite number of them.     

Two-primary algebraic K-theory of two-regular number fields

We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order. Received August 1, 1998