Algebraic Numbers Of Degree Research Articles

A relation between successive approximations and continued fractions for quadratic irrationals: rates of convergence

We find the rates of convergence of sequences of rational convergents for continued fractions of algebraic numbers of degree 2, the quadratic irrationals.

Lower bounds for the heights of solutions of linear equations

Let K be a number eld of degree d. We de ne as usual the eld height HK ( ) of a number ∈ K by HK ( ) = ∏ v∈M (K) max{1; | |dv v } : Here the v-adic valuations | |v are normalized such that on Q they coincide with either the standard absolute value or with the p-adic absolute value | |p having |p|p = p−1. Moreover dv denotes the local degree. The absolute height of then, is de ned by H ( ) = HK ( ) : Lehmer’s conjecture says the following. Let be a nonzero algebraic number of degree D. Then; if is not a root of unity we have

Algebraic Numbers with Integral Power Traces

A generalization of the following theorem will be proved. If α is an algebraic number of degree n over Q and the trace of α i lies in Z for all positive i up to n + n log 2 n, then α is an algebraic integer.

A diophantine problem in harmonic analysis

In the problem session of the Journées Arithmétiques 1989 in Luminy, Bourgain made the followingConjecture. Suppose α is an algebraic number of degree d. Assume that for each n∈ ℕThen there exists a constant c = c(α) with the following property: given any subspace W of dimension ≤ d − 1 of the d-dimensional ℚ-vector space ℚ(α), the set {n|n∈ℕ,αn∈W} contains fewer than c(α) elements.

Representation of numbers by quadratic forms in algebraic number fields

Let K be the totally real field of algebraic numbers of degree n=[k∶2e] with the discriminant D=D(K); t=t(x1, ..., xs) a totally positive quadratic form of the determinant d>0 over the ring σ of integers from the field K; S⩾4. Let be the number of representations over σ of the number m∈σ by the form a complete singular series. It is proved that for given s and n, there exists a constant c such that for N(d)>0 it is not true that for all m∈σ with m totally positive.

The Computational Complexity of Algebraic Numbers

Let $\{ {x_i } \}$ be any sequence approximating an algebraic number $\alpha $ of degree r, and let $x_{i + 1} = \varphi (x_i ,x_{i - 1} , \cdots ,x_{i - d + 1} )$, for some rational function $\varphi $ with integral coefficients. Let M denote the number of multiplications or divisions needed to compute $\varphi $ and let $\bar M$ denote the number of multiplications or divisions, except by constants, needed to compute $\varphi $. Define the multiplicative efficiency measure of $\{ {x_i } \}$ as $E = {{(\log _2 p)} / M}$ or as $\bar E = {{(\log _2 p)} / {\bar M}}$, where p is the order of convergence of $\{ {x_i } \}$. Kung [1] showed that $\bar E \leqq 1$ or equivalently, $\bar M \geqq \log _2 p$. In this paper we show that (i) $\bar M \geqq \log _2 [r(\lceil p \rceil - 1) + 1] - 1$; (ii) if $E = 1$ then $\alpha $ is a rational number; (iii) if $\bar E = 1$ then $\alpha $ is a rational or quadratic irrational number. This settles the question of when the multiplicative efficiency E or $\bar E$ achieves its optimal value of unity. Also, as a consequence of result (i), we show that the maximal efficiency $\bar E$ achievable by algebraic numbers of degree r drops at least as $O[(\log r)^{ - 1} ]$, provided that we only consider sequences $\{ {x_i } \}$ of bounded order of convergence.

Distribution of values of certain classes of additive arithmetic functions in algebraic number fields

An investigation is made of the generalization of a theorem of B. V. Levin and A. S. Fainleib for homothetically extending regions in a certain n-dimensional real space connected with a given field K of algebraic numbers of degree n≥2; the paper also investigates applications of the theorem to the problem of the distribution of real additive functions which are given on a set of ideal numbers and which belong to a wider class than the class H of I. P. Kubilyus.