Conjugate Algebraic Numbers Research Articles

On distribution densities of algebraic points under different height functions

In the article we consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed degree. The distribution is introduced using a height function. We have obtained universal upper and lower bounds of the distribution density of such points using an arbitrary height function. We have shown how from a given joint density function of coefficients of a random polynomial of degree n, one can construct such a height function H that the polynomials q of degree n uniformly chosen under H[q] ≤1 have the same distribution of zeros as the former random polynomial.

On algebraic points of fixed degree and bounded height

We consider the spatial distribution of points, whose coordinates are conjugate algebraic numbers of fixed de- gree and bounded height. In the article the main result of a recent joint work by the author and F. Götze, and D. N. Zaporozhets is extended to the case of arbitrary height functions. We prove an asymptotic formula for the number of such algebraic points lying in a given spatial region. We obtain an explicit expression for the density function of algebraic points under an arbitrary height function.

Joint distribution of conjugate algebraic numbers: A random polynomial approach

We count the algebraic numbers of fixed degree by their w-weighted lp-norm which generalizes the naïve height, the length, the Euclidean and the Bombieri norms. For non-negative integers k,l such that k+2l≤n and a Borel subset B⊂R×C+l denote by Φp,w,k,l(Q,B) the number of ordered (k+l)-tuples in B of conjugate algebraic numbers of degree n and w-weighted lp-norm at most Q. We show thatlimQ→∞⁡Φp,w,k,l(Q,B)Qn+1=Voln+1(Bp,wn+1)2ζ(n+1)∫Bρp,w,k,l(x,z)dxdz, where Voln+1(Bp,wn+1) is the volume of the unit w-weighted lp-ball and ρp,w,k,l will denote the correlation function of k real and l complex zeros of the random polynomial ∑j=1nηjwjzj, where ηj are i.i.d. random variables with density cpe−|t|p for 0<p<∞ and with constant density on [−1,1] for p=∞. If the boundary of B is of Lipschitz type, we also estimate the rate of convergence. We give an explicit formula for ρp,w,k,l, which in the case k+2l=n has a very simple form. To this end, we obtain a general formula for the correlations between real and complex zeros of a random polynomial with arbitrary independent absolutely continuous coefficients.

Roots of unity as quotients of two conjugate algebraic numbers

Roots of unity as quotients of two conjugate algebraic numbers

Distance between conjugate algebraic numbers in clusters

For integers n ≥ 2 and Q> 1, we introduce the class of integer polynomials P (x )= anx n + an−1x n−1 + ··· + a1x + a0 such that

On relations for rings generated by algebraic numbers and their conjugates

Let \(\alpha \) be an algebraic number of degree \(d\) with minimal polynomial \(F \in \mathbb {Z}[X]\), and let \(\mathbb {Z}[\alpha ]\) be the ring generated by \(\alpha \) over \(\mathbb {Z}\). We are interested whether a given number \(\beta \in \mathbb {Q}(\alpha )\) belongs to the ring \(\mathbb {Z}[\alpha ]\) or not. We give a practical computational algorithm to answer this question. Furthermore, we prove that a rational number \(r/t \in \mathbb {Q}\), where \(r \in \mathbb {Z}, t \in \mathbb {N}, \gcd (r, t) = 1\), belongs to the ring \(\mathbb {Z}[\alpha ]\) if and only if the square-free part of its denominator \(t\) divides all the coefficients of the minimal polynomial \(F \in \mathbb {Z}[X]\) except for the constant coefficient \(F(0)\) that must be relatively prime to \(t\), namely \(\gcd (F(0),t)=1\). We also study the question when the equality \(\mathbb {Z}[\alpha ] = \mathbb {Z}[\alpha ']\) for algebraic numbers \(\alpha , \alpha '\) conjugates over \(\mathbb {Q}\) holds. In particular, it is shown that for each \(d \in \mathbb {N}\), there are conjugate algebraic numbers \(\alpha , \alpha '\) of degree \(d\) satisfying \(\mathbb {Q}(\alpha ) = \mathbb {Q}(\alpha ')\) and \(\mathbb {Z}[\alpha ] \ne \mathbb {Z}[\alpha ']\). The question concerning the equality \(\mathbb {Z}[\alpha ]=\mathbb {Z}[\alpha ']\) is answered completely for conjugate quadratic pairs \(\alpha ,\alpha '\) and also for conjugate pairs \(\alpha , \alpha '\) of cubic algebraic integers.

Simultaneous Diophantine approximation in two metrics and the distance between conjugate algebraic numbers in [formula omitted

A lower bound for the number of integer polynomials which simultaneously have “close” complex roots and “close” p -adic roots is obtained.

Distribution of algebraic numbers

Schur studied limits of the arithmetic means A n of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that . We show that A n → 0, and estimate the rate of convergence by generalizing the Erdős–Turán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line. Potential theoretic methods allow us to consider the distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure.

The distribution of close conjugate algebraic numbers

AbstractWe investigate the distribution of real algebraic numbers of a fixed degree that have a close conjugate number, with the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general, it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds obtained by Bugeaud and Mignotte. So far, the results à la Bugeaud and Mignotte have relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. We consider some applications of our main theorem, including generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.

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.

Additive Hilbert's Theorem 90 in the ring of algebraic integers

For a number field K, we give a complete characterization of algebraic numbers which can be expressed by a difference of two K-conjugate algebraic integers. These turn out to be the algebraic integers whose Galois group contains an element, acting as a cycle on some collection of conjugates which sum to zero. Hence there are no algebraic integers which can be written as a difference of two conjugate algebraic numbers but cannot be written as a difference of two conjugate algebraic integers. A generalization of the construction to a commutative ring is also given. Furthermore, we show that for n ⩾_ 3 there exist algebraic integers which can be written as a linear form in n K-conjugate algebraic numbers but cannot be written by the same linear form in K-conjugate algebraic integers.

Conjugate algebraic numbers close to a symmetric set

A new proof is presented for the Motzkin theorem saying that if a set consists of d− 1 complex points and is symmetric relative to the real axis, then there exists a monic, irreducible, and integral polynomial of degree d whose roots are as close to each of these d−1 points as we wish. Unlike the earlier proofs, the new proof is efficient, i.e., it gives both an explicit construction of the polynomial in question and the location of its dth root. §

SIMULTANEOUS APPROXIMATION BY CONJUGATE ALGEBRAIC NUMBERS IN FIELDS OF TRANSCENDENCE DEGREE ONE

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, …, ξt by conjugate algebraic numbers of bounded degree over ℚ, provided that the given transcendental numbers ξ1, …, ξt generate over ℚ a field of transcendence degree one. We provide sharper estimates for example when ξ1, …, ξt form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at ξ1, …, ξt.

A Gel'fond type criterion in degree two

We establish a criterion for a complex number to be algebraic over Q of degree at most two. It requires that, for any sufficiently large real number X, there exists a non-zero polynomial with integral coefficients, of degree at most two and height at most X, whose absolute value at that complex number is at most (1/4)X^{-(3+sqrt{5})/2}. We show that the exponent (3+sqrt{5})/2 in this condition is optimal, and deduce from this criterion a result of simultaneous approximation of a real number by conjugate algebraic numbers.

Additive relations with conjugate algebraic numbers

Additive relations with conjugate algebraic numbers

Additive and multiplicative relations connecting conjugate algebraic numbers

We investigate the problem of finding for which sets of integers a 1,…, a k either of the equations Σ i = 1 k a i α i = 0 or Π i = 1 k α i a i = 1 has a non-trivial solution in (not necessarily distinct) conjugate algebraic numbers α 1,…, α k . The problem turns out to be connected with the existence of certain latin squares having zero determinant.

Conjugate algebraic numbers on conics

Conjugate algebraic numbers on conics

Conjugate algebraic numbers on circles

Conjugate algebraic numbers on circles