Real Algebraic Integers Research Articles

On the distribution of eigenvalues of increasing trees

We prove that the multiplicity of a fixed eigenvalue α in a random recursive tree on n vertices satisfies a central limit theorem with mean and variance asymptotically equal to μαn and σα2n respectively. It is also shown that μα and σα2 are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue 0, the constants μ0 and σ02 can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.

Totally real algebraic integers in short intervals, Jacobi polynomials, and unicritical families in arithmetic dynamics

AbstractWe classify all postcritically finite unicritical polynomials defined over the maximal totally real algebraic extension of . Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion that allows one to give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than 4.

SYMMETRICAL MATRICES CHARACTERISTIC OF INTEGERS WITH INTEGER EIGEN VALUE

Estes (1992) stated that the set of eigen values from symmetrical matrices of Z is a set of totally real algebraic integers. Estes was not able to ensure that eigen values of a symmetrical matrices are integers. Mckee and Smyth (2007) observed more about the eigen value of symmetrical integer matrices. James and Chris proved that symmetrical integer matrices have eigen value with interval ranging in [-2,2]. Contrary to that, Martin and Wong (2009), stated that almost all integer matrices have no integer eigen value. Previous studies that could not show the characteristic of the eigen value made Cao and Koyunco studied and tried to determine the characteristic of symmetrical integer matrices for rank 2 and rank 3. The result shows that they have integer eigen value. In accordance to Cao and Kuyonco study, this article elaborates the characteristic of a symmetrical integer matrices for rank 4, and 5 to show the characteristic of a symmetrical integer matrices with integer eigen value for rank 1, 2, 3 and for rank 4 and 5.

AF-algebras with lattice-ordered K0: Logic and computation

Following Bratteli's original definition, an AF-algebra A is the norm closure of the union of an ascending sequence of finite-dimensional C⁎-algebras, all with the same unit. Elliott proved that the unital dimension (Grothendieck) group K0(A) uniquely determines A up to isomorphism. AF-algebras with lattice-ordered K0, (AFℓ-algebras) have a preeminent role in the AF-algebraic literature. The Elliott classifier E(A) of any AFℓ-algebra A—i.e., the unit interval of K0(A), or equivalently, the countable set of Murray-von Neumann equivalence classes of projections of A—has the structure of a countable MV-algebra. Every countable MV-algebra arises as E(A) for some AFℓ-algebra A. Since E(A) is the Lindenbaum algebra of some countable set of formulas in Łukasiewicz logic Ł∞, every Ł∞-formula ϕ naturally codes an equivalence class ϕA of projections in A. The deductive-algorithmic machinery of Ł∞ can be thus applied to decide, among many others, the following basic problems, with ⪯ the Murray-von Neumann order:Is ϕA⪯ψA?IsϕA={0}? Is ϕA central ? For any AFℓ-algebra A we construct mutual polytime reductions of the first two problems. The third problem is polytime reducible to both problems. If E(A) is finitely generated and A either has no quotient isomorphic to C or is simple, then the first two problems are polytime reducible to the third one. The complexity of these problems is polytime for many AFℓ-algebras in the literature, including the Behncke-Leptin algebras Am,n, the Farey AFℓ-algebra M1 and all its principal quotients, the CAR algebra and Glimm's universal UHF algebra, and every Effros-Shen algebra Fθ for θ∈[0,1]∖Q a real algebraic integer, or for θ=1/e. For any fixed n=1,2,… we consider the class Kn of AFℓ-algebras presented by generators X1,…,Xn and relations θ1=0,…,θm=0, for an arbitrary finite set Θ={θ1,…,θm} of Ł∞-formulas in the variables X1,…,Xn. We prove that all decision problems studied in this paper are polytime decidable for Kn—uniformly over the presentation Θ.

Monogenic Pisot and Anti-Pisot Polynomials

A Pisot number is a real algebraic integer $\alpha \gt 1$ such that all of its Galois conjugates, other than $\alpha$ itself, lie inside the unit circle. An anti-Pisot number is a real algebraic integer $\alpha \gt 1$, such that exactly one Galois conjugate of $\alpha$ lies inside the unit circle, and $\alpha$ has at least one Galois conjugate, other than $\alpha$ itself, outside the unit circle. We call the minimal polynomials of these algebraic integers, respectively, Pisot and anti-Pisot polynomials. In this article, we find infinite families of Pisot (anti-Pisot) polynomials $f(x)$ such that $\{ 1, \alpha, \alpha^2, \ldots, \alpha^{n-1} \}$ is a basis for the ring of integers of $\mathbb{Q}(\alpha)$, where $\alpha$ is a Pisot (anti-Pisot) number and $\deg(f) = n$, for certain $n$. We refer to these polynomials as monogenic Pisot (anti-Pisot) polynomials.

Mahler measures of Pisot and Salem type numbers

In this paper we consider the set of Mahler measures of generalized Pisot and Salem numbers and give a characterization of such numbers in terms of their minimal polynomials. We show that every real quadratic algebraic integer β can be expressed by the difference of the Mahler measure of a generalized Pisot number and a Pisot number in ℚ(β).

Linear relations with conjugates of a Salem number

In this paper we consider linear relations with conjugates of a Salem number $\alpha$. We show that every such a relation arises from a linear relation between conjugates of the corresponding totally real algebraic integer $\alpha+1/\alpha$. 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 $6$.

Positive semigroups and generalized Frobenius numbers over totally real number fields

Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors satisfying some special conditions. We also use a result of Borosh and Treybig to obtain bounds on the size of representations and number of elements of bounded height in such positive semigroups of totally real algebraic integers.

Julia Robinson numbers and arithmetical dynamic of quadratic polynomials

For rings $\mathcal{O}_K$ of totally real algebraic integers, J. Robinson defined a set which is always $\{+\infty\}$ or of the form $[\lambda,+\infty)$ or $(\lambda,+\infty)$ for some real number $\lambda\ge4$. All known examples give either $\{+\infty\}$ or $[4,+\infty)$. In this paper, we construct infinitely many fields such that the set is an interval, but not equal to $[4,+\infty)$.

Distribution of real algebraic integers

In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their naïve height tends to infinity. For an arbitrary interval I⊂R and sufficiently large Q>0, we obtain an asymptotic formula for the number of algebraic integers α∈I of fixed degree n and naïve height H(α)≤Q. In particular, we show that the real algebraic integers of degree n, with their height growing, tend to be distributed like the real algebraic numbers of degree n−1. However, we reveal two symmetric “plateaux”, where the distribution of real algebraic integers statistically resembles the rational integers.

Algebraic numbers with elements of small height

AbstractIn this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers.

Pseudorandom number generator based on the Bernoulli map on cubic algebraic integers.

We develop a pseudorandom bit generator using chaotic true orbits of the Bernoulli map on real cubic algebraic integers having complex conjugates. Although this generator has a high computational cost, it exactly simulates the Bernoulli map that can generate ideal random binary sequences. In particular, we clarify a seed selection method that can select initial points (i.e., seeds) without bias and can avoid overlaps in latter parts of the pseudorandom binary sequences derived from them. Moreover, in order to evaluate the memory usage of our generator, we give upper bounds concerning the growth of the representation of points on a true orbit. We also report results of a large variety of tests indicating that the generated pseudorandom sequences have good statistical properties as well as an advantage over one of the most popular generators at this point, the Mersenne Twister MT19937.

Form class groups for extended ring class fields

Let L be an extended ring class field of an imaginary quadratic field K other than Q(−1) and Q(−3). We show that there is a form class group induced from a congruence subgroup which describes the Galois group of L over K in a concrete way. We also construct a primitive generator of L over K as a real algebraic integer which can be applied to certain quadratic Diophantine equations.

Abelian spiders and real cyclotomic integers

If Γ \Gamma is a finite graph, then the largest eigenvalue λ \lambda of the adjacency matrix of Γ \Gamma is a totally real algebraic integer ( λ \lambda is the Perron-Frobenius eigenvalue of Γ \Gamma ). We say that Γ \Gamma is abelian if the field generated by λ 2 \lambda ^2 is abelian. Given a fixed graph Γ \Gamma and a fixed set of vertices of Γ \Gamma , we define a spider graph to be a graph obtained by attaching to each of the chosen vertices of Γ \Gamma some 2 2 -valent trees of finite length. The main result is that only finitely many of the corresponding spider graphs are both abelian and not Dynkin diagrams, and that all such spiders can be effectively enumerated; this generalizes a previous result of Calegari, Morrison, and Snyder. The main theorem has applications to the classification of finite index subfactors. We also prove that the set of Salem numbers of “abelian type” is discrete.

Pisot unit generators in number fields

Pisot numbers are real algebraic integers bigger than 1, whose other conjugates all have modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called complex Pisot numbers and we solve it in all number fields that admit such a generator, in particular this includes all fields without CM.

Intersections of Multiplicative Translates of 3-Adic Cantor Sets II: Two Infinite Families

ABSTRACTThis article studies the structure of finite intersections of general multiplicative translates for integers 1 ⩽ M1 < M2 < ⋅⋅⋅ < Mn, in which denotes the 3-adic Cantor set (of 3-adic integers whose expansions omit the digit 2), which has Hausdorff dimension log 32 ≈ 0.630929. This study was motivated by questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. The exceptional set is defined to be the set of all elements of whose forward orbits under this action intersect the 3-adic Cantor set infinitely many times. It is conjectured that it has Hausdorff dimension 0. An earlier article showed that upper bounds on the Hausdorff dimension of the exceptional set can be extracted from knowing Hausdorff dimensions of sets of the kind above, in cases where all Mi are powers of 2. These intersection sets were shown to be fractals whose points have 3-adic expansions describable by labeled paths in a finite automaton, whose Hausdorff dimension is exactly computable and is of the form log 3(β), where β is a real algebraic integer. It gave algorithms for determination of the automaton, and computed examples showing that the dependence of the automaton and the value β on the parameters (M1, …, Mn) is complicated. The present article studies two new infinite families of examples, illustrating interesting behavior of the automata and of the Hausdorff dimension of the associated fractals. One family has associated automata whose directed graph has a nested sequence of strongly connected components of arbitrarily large depth. The second family leads to an improved upper bound for the Hausdorff dimension of the exceptional set of log 3φ ≈ 0.438018, where φ denotes the Golden ratio.

On the growth rate of ideal Coxeter groups in hyperbolic 3-space

We study the set $\mathcal{G}$ of growth rates of ideal Coxeter groups in hyperbolic 3-space; this set consists of real algebraic integers greater than 1. We show that (1) $\mathcal{G}$ is unbounded above while it has the minimum, (2) any element of $\mathcal{G}$ is a Perron number, and (3) growth rates of ideal Coxeter groups with $n$ generators are located in the closed interval $[n-3, n-1]$.

Primitive generators of certain class fields

We find primitive generators of certain class fields of imaginary quadratic fields as real algebraic integers which arise from the study of some quadratic Diophantine equations.

Every totally real algebraic integer is a tree eigenvalue

Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite graph is a deep and remarkable result, conjectured forty years ago by Hoffman, and proved seventeen years later by Estes. This short paper provides an independent and elementary proof of a stronger statement, namely that the graph may actually be chosen to be a tree. As a by-product, our result implies that the atoms of the limiting spectrum of n×n symmetric matrices with independent Bernoulli (cn) entries are exactly the totally real algebraic integers. This settles an open problem raised by Ben Arous (2010).

Pisot Numbers from {0, 1}-Polynomials

AbstractA Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.