Multiplicative relations with conjugate algebraic numbers

We study what algebraic numbers can be represented by a product of algebraic numbers conjugate over a fixed number field K in fixed integer powers. The problem is nontrivial if the sum of these integer powers is equal to zero. The norm of such a number over K must be a root of unity. We show that there are infinitely many algebraic numbers whose norm over K is a root of unity and which cannot be represented by such a product. Conversely, every algebraic number can be expressed by every sufficiently long product in algebraic numbers conjugate over K. We also construct nonsymmetric algebraic numbers, i.e., algebraic numbers such that no elements of the corresponding Galois group acting on the full set of their conjugates form a Latin square. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 890–900, July, 2007.     

Additive and multiplicative relations connecting conjugate algebraic numbers

We investigate the problem of finding for which sets of integers a₁,…, a_k either of the equations Σ_{i = 1}^k a_iα_i = 0 or Π_{i = 1}^k α_i^a_i = 1 has a non-trivial solution in (not necessarily distinct) conjugate algebraic numbers α₁,…, α_k. The problem turns out to be connected with the existence of certain latin squares having zero determinant.     

The dimension of fields and algebraic K-theory

In this paper, we consider certain K-theoretic modifications of the condition C_i of Lang. We propose a conjecture which relates these conditions to the cohomological dimension of fields. Partial solutions are given for local and global fields.     

On algebraic approximations of certain algebraic numbers

Using hypergeometric functions and the Thue-Siegel method we give an effective improvement of Liouville's approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalities

|BX4−AX3Y−6BX2Y2+AXY3+BY4|?N     

Simplifying method for algebraic approximation of certain algebraic numbers

A new simple method for approximating certain algebraic numbers is developed. By applying this method, an effective upper bound is derived for the integral solutions of the quartic Thue equation with two parameters $tx^4 - 4sx^3 y - 6tx^2 y^2 + 4sxy^3 + ty^4 = N$ , where s > 32t ³. As an application, Ljunggren’s equation is solved in an elementary way.     

Cohomological dimension of certain algebraic varieties

Let be an ideal of a commutative Noetherian ring . For finitely generated -modules and with , it is shown that . Let be a finitely generated module over a local ring such that . Using the above result and the notion of connectedness dimension, it is proved that Here denotes the connectedness dimension of the topological space . Finally, as a consequence of this inequality, two previously known generalizations of Faltings' connectedness theorem are improved.

     

The existence of algebraic surfaces with preassigned Chern numbers

To Cao Xi-Hua on his 70th Birthday     

The algebraic uniqueness of the addition of natural numbers

As is known, multiplication is aniteration of addition within the natural numbers. We show that addition is essentially theonly associative binary operation on natural numbers whose iteration is commutative or associative. This result, in which a set determines a structure, is in a sense a converse of the classical one due to Pontrjagin [3] and Kolmogorov [1], in which certain algebraic and topological structures determine the fields of real and complex numbers and of quaternions.     

On the question of the algebraic independence of algebraic powers of algebraic numbers

We obtain results showing that transcendental numbers of the form , wherea0, 1, is irrational, anda and are algebraic numbers, cannot be expressed algebraically in terms of two of the numbers. The proof is carried out by A. O. Gel'fond's method.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 635–644, June, 1972.     

Sums of heights of algebraic numbers

For , we consider the set . The polynomials are in , with only mild restrictions, and is the Weil height of . We show that this set is dense in for some effectively computable limit point .

     

Rational and algebraic approximations of algebraic numbers and their application

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and ε =x ₀ denotes the fundamental solution ofx ²?Dy ²=?1, thenx ²+1=Dy ⁴ is solvable if and only ify ₀=A ², where A is an integer. Moreover, if ε>64, thenx ²+1=Dy ⁴ has at most one positive integral solution (x, y).     

Representations of algebraic numbers by matrices

Two examples are given, showing the connections between the representations of algebraic numbers of a field by rational symmetric matrices, and the representations of integers of a field by symmetric integer-valued matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 46, pp. 89–91, 1974.