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 complex numbers are called algebraic integers "totally" in a compact subset, which means algebraic integers all conjugates of which belong to a same given compact subset of the complex plane.
The study of such algebraic integers helps to understand the eigenvalues of the Frobenius endomorphism, especially their distribution. In this paper, we will study the following question : under which conditions a compact subset of the complex plane has a finite or infinite number of algebraic integers "totally" in it ? The problem can be studied in light of the notion of capacity of a compact subset, which comes from potential theory. In this paper, we will present the theory of capacity and some theorems (Fekete, Szegö, Robinson) derived from it that partially answer the question: in the case of a union of real segments, when the capacity is smaller (resp. larger) than 1, it contains a finite (resp. infinite) number of algebraic integers totally in it. For instance, for real line segments, the limit length is 4.
This paper is written as part of a collective project conducted in École Polytechnique (France). It is aimed towards undergraduate audience in mathematics, with basic knowledge in algebra, topology, analysis, and dwells into a modern topic of research.
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 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 vecto…
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…
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.
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…
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.
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…
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 present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first first family $\mathcal{P}$; and its sum, the Thue-Morse sequence, belongs to the second family $\mathcal{G}$. The family $\mathcal{G}$ contains all the iterated sums of sequences from the $\mathcal{P}$ and more.
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.
Given a subvariety $V$ of the complex algebraic torus ${\mathbb G}_{\rm m}^n$ defined by polynomials of total degree at most $d$ and a power map $ϕ: {\mathbb G}_{\rm m}^n \to {\mathbb G}_{\rm m}^n$, the points ${\bf x}$ whose forward orbits ${\mathcal O}_ϕ({\bf x})$ belong to $V$ form its {\em stable} subvariety $S(V,ϕ)$. The main result of the paper provides an upper bound $T=T(n,d,ϕ)$ for the number of iterations of the power map $ϕ$ required to ``cut off'' the points of $V$ that do not belong to $S$.
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 $D$.
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…
A computably presented algebraic field $F$ has a \emph{splitting algorithm} if it is decidable which polynomials in $F[X]$ are irreducible there. We prove that such a field is computably categorical iff it is decidable which pairs of elements of $F$ belong to the same orbit under automorphisms. We also show that this criterion is equivalent to the relative computable categoricity of $F$.
As is well-known, the compact groups Spin(7) and SO(7) both have a single conjugacy class of compact subgroups of exceptional type G_2. We first show that if H is a subgroup of Spin(7), and if each element of H is conjugate to some element of G_2, then H itself is conjugate to a subgroup of G_2. The analogous statement for SO(7) turns out be false, and our main result is a classification of all the exceptions. They are the following groups, embedded in each case in SO(7) in a very specific way: GL_2(Z/3Z), SL_2(Z/3Z), Z/4Z x Z/2Z, as well as the nonabelian subgroups of GO_2(C) with compact clo…
