We find sharp upper and lower bounds for the degree of an algebraic number in terms of the $Q$-dimension of the space spanned by its conjugates. For all but seven nonnegative integers $n$ the largest degree of an algebraic number whose conjugates span a vector space of dimension $n$ is equal to $2^n n!$. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of $GL_n(Q)$; this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4.
We extend our results in two directions. We consider the problem when $Q$ is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension of $Q$. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number.
In this paper, we present a result on using algebraic conjugates to form a sequence of approximations to an algebraic number, and in this way obtain effective irrationality measures for related algebraic numbers. From this result, we are able to generalise Thue's Fundamentaltheorem.
Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree n, and 2) by all but one conjugates of algebraic integers of degree n+1.
We investigate the distribution of real algebraic numbers of a fixed degree having a close conjugate number, 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 of Bugeaud and Mignotte. So far the results a la Bugeaud and Mignotte relied on finding explicit fami…
We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers xi_1,...,xi_t by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers xi_1,...,xi_t generate over Q a field of transcendence degree one. We provide sharper estimates for example when xi_1,...,xi_t 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 crite…
Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $ξ$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $ξ$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $ξ$ that is either transcendental or algebraic of sufficiently large degree…
Let $φ:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable function on an interval $J\subset\mathbb{R}$ and let $\boldsymbolα=(α_1,α_2)$ be a point with algebraic conjugate integer coordinates of degree $\leq n$ and of height $\leq Q$. Denote by $\tilde{M}^n_φ(Q,γ, J)$ the set of points $\boldsymbolα$ such that $|φ(α_1)-α_2|\leq c_1 Q^{-γ}$. In this paper we show that for a real $0<γ<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, which are independent of $Q$, such that $c_2\cdot Q^{n-γ}<# \tilde{M}^n_φ(Q,γ, J)< c_3\cdot Q^{n-γ}$.
The beta-conjugates of a base of numeration $β> 1$, $β$ being a Parry number, were introduced by Boyd, in the context of the Rényi-Parry dynamics of numeration system and the beta-transformation. These beta-conjugates are canonically associated with $β$. Let $β> 1$ be a real algebraic number. A more general definition of the beta-conjugates of $β$ is introduced in terms of the Parry Upper function $f_β(z)$ of the beta-transformation. We introduce the concept of a germ of curve at $(0,1/β) \in \mathbb{C}^{2}$ associated with $f_β(z)$ and the reciprocal of the minimal polynomial of $β$. This ger…
We show that for any sufficiently large integer $Q$ and a real $0\leqλ\leq\frac34$ there exists a value $c(n,f,J)>0$ such that all strips $L(Q,λ)=\{(x,y):|y-f(x)|<Q^{-λ}, x\in J=[a,b]\}$ contain at least $c(n, f, J)Q^{n+1-λ}$ points $\barγ=(α,β)$ with algebraically conjugate coordinates. We consider points $\barγ$ such that the minimal polynomial $P(x)$ of $α,β$ is of degree $°P\leq n,\ n\ge 2$, and height $H(P)\leq Q$. The proof is based on a metric theorem on the measure of the set of vectors $(x,y)$ lying in a rectangle $Π$ of dimensions $Q^{-s_1}\times Q^{-s_2}$ with $|P(x)|, |P(y)|$ bound…
In an earlier work, the first author and Petsche solved an energy minimization problem for local fields and used the result to obtain lower bounds on the height of algebraic numbers all whose conjugates lie in various local fields, such as totally real and totally p-adic numbers. In this paper, we extend these techniques and solve the corresponding minimization programs for real intervals and p-adic discs, obtaining several new lower bounds for the height of algebraic numbers all of whose conjugates lie in such sets.
We consider conjugacy of integral matrices by elements in $\text{GL}_{n}(R)$ for certain rings $R$ with subring $\mathbb{Z}$. We note that a Hasse principal does not hold in the context of matrix conjugacy because matrices which are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all $p$ are not necessarily $\text{GL}_{n}(\mathbb{Z})$-conjugate. By a theorem of Guralnick, we know that integral $n \times n$ matrices are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all primes $p$ if and only if they are conjugate by an element in $\text{GL}_{n}(E)$ for some algebraic integral extension $E$ of $\mathb…
In this paper we consider linear relations with conjugates of a Salem number $α$. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer $α+1/α$. It is also shown that the smallest degree of a Salem number with a nontrivial relation between its conjugates is $8$, whereas the smallest length of a nontrivial linear relation between the conjugates of a Salem number is $6$.
For $B\subset\mathbb{R}^k$ denote by $Φ_k(Q;B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\leq n$ and naive height $\leq Q$. We show that $$ Φ_k(Q;B) = \frac{(2Q)^{n+1}}{2ζ(n+1)} \int_{B} ρ_k(\mathbf{x})\,d\mathbf{x} + O\left(Q^n\right),\quad Q\to \infty, $$ where the function $ρ_k$ will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term.
We study periodic representations in number systems with an algebraic base $β$ (not a rational integer). We show that if $β$ has no Galois conjugate on the unit circle, then there exists a finite integer alphabet $\mathcal A$ such that every element of $\mathbb Q(β)$ admits an eventually periodic representation with base $β$ and digits in $\mathcal A$.
A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed. Some generalisations are given too, though not to Mahler measure of polynomials in more than one variable.
Schur introduced the problem on the smallest limit point for the arithmetic means of totally positive conjugate algebraic integers. This area was developed further by Siegel, Smyth and others. We consider several generalizations of the problem that include questions on the smallest limit points of symmetric means. The key tool used in the study is the asymptotic distribution of algebraic numbers understood via the weak* limits of their counting measures. We establish interesting properties of the limiting measures, and find the smallest limit points of symmetric means for totally positive alge…
Let $α$ be an algebraic integer of degree $d$, which is reciprocal. The house of $α$ is the largest modulus of its conjugates. We compute the minimum of the houses of all reciprocal algebraic integers of degree $d$ which are not roots of unity, say $\mathrm{mr}(d)$, for $d$ at most 34. We prove several lemmata and use them to avoid unnecessary calculations. The computations suggest several conjectures. The direct consequence of the last one is the conjecture of Schinzel and Zassenhaus. We demonstrate the utility of $d$-th power of the house of $α$.
We extend a result of Hančl, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{α_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$ \limsup_{n \rightarrow \infty} \vertα_n\vert^{\frac{1}{Dd^{n-1} \prod_{i=1}^{n-2}(Dd^i + 1)}} = \infty $$ implies that the continued fraction $α= [0;α_1, α_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.
The study of Frobenius endomorphism provides numerous information about its corresponding Abelian variety. To understand the action of the Frobenius endomorphism, one may be interested in its eigenvalues. According to Weil's third conjecture ("Riemann hypothesis over finite fields"), they all have absolute value less than or equal to $2g\sqrt{p}$. Thus, the eigenvalues of the Frobenius endomorphism all belong to the same compact subset of the complex plane, and are roots of the same monic polynomial with integer coefficients (the characteristic polynomial of the Frobenius endomorphism). Such c…
Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field extension of $k$ of degree bounded independently of the subgroups. Moreover, if $k$ is a real number field, we show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over the field of real numbers $\mathbb{R}$ are actually conjugate over a finite real extension of $k$ of degree bounded independently of the subgroups.
We show that conjugation by an automorphism of the complex numbers (as an abstract field) may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups.
Let $α$ be an algebraic integer of degree $d$, which is reciprocal. The house of $α$ is the largest modulus of its conjugates. We proved that $d$-th power of the house of reciprocal $α$ has a limit point. We presented a property of antireciprocal hexanomials. We compute the minimum of the houses of all reciprocal algebraic integers of degree $d$ having the minimal polynomial which is a factor of a $D$-th degree reciprocal or antireciprocal polynomial with at most eight monomials, say $\mathrm{mr}(d)$, for $d$ at most 180, $D\le 1.5d$ and $D\le 210$. We show that it is not necessary to take int…
Let $1<q<2$ and \[ Λ(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $Λ(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $Λ(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $α$ such that $q|α|<1$, then $Λ(q)$ is dense in $\mathbb{R}$.
We count the algebraic numbers of fixed degree by their $\mathbf{w}$-weighted $l_p$-norm which generalizes the naïve height, the length, the Euclidean and the Bombieri norms. For non-negative integers $k,l$ such that $k+2l\leq n$ and a Borel subset $B\subset \mathbb{R}\times\mathbb{C}_+^l$ denote by $Φ_{p,\mathbf{w},k,l}(Q,B)$ the number of ordered $(k+l)$-tuples in $B$ of conjugate algebraic numbers of degree $n$ and $\mathbf{w}$-weighted $l_p$-norm at most $ Q$. We show that $$ \lim_{ Q\to\infty}\frac{Φ_{p,\mathbf{w},k,l}( Q,B)}{ Q^{n+1}}=\frac{\mathrm{Vol}_{n+1}(\mathbb{B}_{p,\mathbf{w}}^{n…
We count algebraic numbers of fixed degree $d$ and fixed (absolute multiplicative Weil) height $\mathcal{H}$ with precisely $k$ conjugates that lie inside the open unit disk. We also count the number of values up to $\mathcal{H}$ that the height assumes on algebraic numbers of degree $d$ with precisely $k$ conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if $k \in \{0,d\}$ or…
For any number field F, call a cusp form πon GL(2)/F {\it special icosahedral}, or just s-icosahedral for short, if πis not solvable polyhedral, and for a suitable "conjugate" cusp form π' on GL(2)/F, sym^3(π) is isomorphic to sym^3(π'), and the symmetric fifth power L-series of πequals the Rankin-Selberg L-function L(s, sym^2(π') x π) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let πbe s-icosahedral (of trivial central character). Then π_f is algebraic without local components of Steinberg type, π_\infty is of Galois type, and π_v i…
Let $K$ be an algebraically closed field of characteristic 0 and $f\in K[t]$ a polynomial of degree $d\geq 2$. There exists a local conjugacy $ψ_f(t)\in tK[[1/t]]$ such that $ψ_f(t^d)=f(ψ_f(t))$. It has been known that $ψ_f$ is transcendental over $K(t)$ if $f$ is not conjugate to $t^d$ or a constant multiple of the Chebyshev polynomial. In this paper, we study the algebraic independence of $ψ_{f_1}$,\ldots,$ψ_{f_n}$ using a recent result of Medvedev-Scanlon. Related questions in transcendental number theory and canonical heights in arithmetic dynamics are also discussed.
