Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers

Concentration and equi-distribution, near the unit circle, in Solomyak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number $\beta$ are characterized by means of the Erd\H{o}s-Tur\'an approach, and its improvements by Mignotte and Amoroso, applied to the analytical function $f_{\beta}(z) = -1 + \sum_{i \geq 1} t_i z^i$ associated with the R\'enyi $\beta$-expansion $d_{\beta}(1)= 0.t_1 t_2 ...$ of unity. Mignotte's discrepancy function requires the knowledge of the factorization of the Parry polynomial of $\beta$. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclotomic, reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of Mignotte's discrepancy function which arises from the beta-conjugates of $\beta$ which are roots of cyclotomic factors is linked to the Riemann hypothesis, following Amoroso. An equidistribution limit theorem, following Bilu's theorem, is formulated for the concentration phenomenon of conjugates of Parry numbers near the unit circle. Parry numbers are Perron numbers. Open problems on non-Parry Perron numbers are mentioned in the context of the existence of non-unique factorizations of elements of number fields into irreducible Perron numbers (Lind).Comment: 33 page

Large Galois groups with applications to Zariski density

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this end we describe efficient algorithms to check if the Galois group of a polynomial $p$ with integer coefficients is "generic" (which, for arbitrary polynomials of degree $n$ means the full symmetric group $S_n,$ while for reciprocal polynomials of degree $2n$ it means the hyperoctahedral group $C_2 \wr S_n.$). We give efficient algorithms to verify that a polynomial has Galois group $S_n,$ and that a reciprocal polynomial has Galois group $C_2 \wr S_n.$ We show how these algorithms give efficient algorithms to check if a set of matrices $\mathcal{G}$ in $\mathop{SL}(n, \mathbb{Z})$ or $\mathop{Sp}(2n, \mathbb{Z})$ generate a \emph{Zariski dense} subgroup. The complexity of doing this in$\mathop{SL}(n, \mathbb{Z})$ is of order $O(n^4 \log n \log \|\mathcal{G}\|)\log \epsilon$ and in $\mathop{Sp}(2n, \mathbb{Z})$ the complexity is of order $O(n^8 \log n\log \|\mathcal{G}\|)\log \epsilon$ In general semisimple groups we show that Zariski density can be confirmed or denied in time of order $O(n^14 \log \|\mathcal{G}\|\log \epsilon),$ where $\epsilon$ is the probability of a wrong "NO" answer, while $\|\mathcal{G}\|$ is the measure of complexity of the input (the maximum of the Frobenius norms of the generating matrices). The algorithms work essentially without change over algebraic number fields, and in other semi-simple groups. However, we restrict to the case of the special linear and symplectic groups and rational coefficients in the interest of clarity.Comment: 25 page

Anosov Automorphisms of Nilpotent Lie Algebras

Each matrix A in GL_n(Z) naturally defines an automorphism f of the free r-step nilpotent Lie algebra on n generators. We study the relationship between the matrix A and the eigenvalues and rational invariant subspaces for f. We give applications to the study of Anosov automorphisms.Comment: 39 page

Roots with common tails

Some cubic polynomials over the integers have three distinct real roots with continued fractions that all have the same common tail. We characterize the polynomials for which this happens, and then investigate the situation for other polynomials of low degree.Comment: 13 pages, no figure

Generating Ray Class Fields of Real Quadratic Fields via Complex Equiangular Lines

For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in $\mathbb{C}^d$ (known in quantum information as symmetric informationally complete measurements or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary real quadratic $K$, and we summarise this in a conjecture. There are indications [19,20] that the logarithms of these canonical units are related to the values of $L$-functions associated to the extensions, following the programme laid out in the Stark Conjectures.Comment: 21 pages. v3 is published version to appear in Acta Arithmetic

Enumeration of a special class of irreducible polynomials in characteristic 2

A-polynomials were introduced by Meyn and play an important role in the iterative construction of high degree self-reciprocal irreducible polynomials over the field F_2, since they constitute the starting point of the iteration. The exact number of A-polynomials of each degree was given by Niederreiter. Kyuregyan extended the construction of Meyn to arbitrary even finite fields. We relate the A-polynomials in this more general setting to inert places in a certain extension of elliptic function fields and obtain an explicit counting formula for their number. In particular, we are able to show that, except for an isolated exception, there exist A-polynomials of every degree.Comment: To appear in Acta Arithmetica. Comments most welcom

Galois conjugates of entropies of real unimodal maps

We investigate the set of Galois conjugates of growth rates of superattracting real quadratic polynomials, following W. Thurston. In particular, we prove that the closure of this set is path-connected and locally connected.Comment: 28 pages, 6 figures. Section 4 modified, introduction and section 7 expande

Vector space of DNA genomic sequences on a Genetic Code Galois Field

A new N-dimensional vector space of DNA sequences over the Galois field of the 64 codons (GF(64)) was recently presented. Now, in order to include deletions and insertions (indel mutations), we have defined a new Galois field over the set of elements X1X2X3 (C125), where Xi belong to {O, A, C, G, C}. We have called this set, the extended triplet set and the elements X1X2X3, the extended triplets. The order of the bases is derived from the Z64-algebra of the genetic code -recently published-. Starting from the natural bijection phi: GF(5^3)-> C125 between the polynomial representation of elements from GF(5^3) and the elements X1X2X3, a novel Galois field over the set of elements X1X2X3 is defined. Taking the polynomial coefficients a0, a1, a2 belong to GF(5) and the bijective function f: GF(5) ->{O, A, C, G, C}, where f(0) = O, f(1) = A, f(2) = C, f(3) = G, f(4) = U, bijection phi is induced such that phi(a0 + a1x + a2x^2) = (f(a1), f(a2), f(a0)) = (X1X2X3). Next, by means of the bijection phi we define sum "+" and product "*" operations in the set of codons C125, in such a way that the resultant field (C125, +, *) turns isomorphic to the Galois Field GF(5^3). This field allows the definition of a novel N-dimensional vector space (S) over the field GF (5^3) on the set of all 125^N sequences of extended triplets in which all possible DNA sequence alignments of length N are included. Here the "classical gap" produced by alignment algorithms corresponds to the neutral element "O". It is verified that the homologous (generalized) recombination between two homologous DNA duplexes involving a reciprocal exchange of DNA sequences -e.g. between two chromosomes that carry the same genetic loci- algebraically corresponds to the action of two automorphism pairs (or two translation pairs) over two paired DNA duplexes.Comment: 10 pages, 2 figure

An asymptotic formula for the number of irreducible transformation shift registers

We consider the problem of enumerating the number of irreducible transformation shift registers. We give an asymptotic formula for the number of irreducible transformation shift registers in some special cases. Moreover, we derive a short proof for the exact number of irreducible transformation shift registers of order two using a recent generalization of a theorem of Carlitz.Comment: 14 page

The GC-content of a family of cyclic codes with applications to DNA-codes

Given a prime power $q$ and a positive integer $r>1$ we say that a cyclic code of length $n$, $C\subseteq F_{q^r}^n$, is Galois supplemented if for any non-trivial element $\sigma$ in the Galois group of the extension $F_{q^r}/ F_q$, $C+C^\sigma= F_{q^r}^n$, where $C^\sigma=\{(x_1^\sigma,\dots,x_n^\sigma)\mid (x_1,\dots,x_n)\in C\}$. This family includes the quadratic-residue (QR) codes over $F_{q^2}$. Some important properties QR-codes are then extended to Galois supplemented codes and a new one is also considered, which is actually the motivation for the introduction of this family of codes: in a Galois supplemented code we can explicitly count the number of words that have a fixed number of coordinates in $F_q$. In connection with DNA-codes the number of coordinates of a word in $F_4^n$ that lie in $F_2$ is sometimes referred to as the $GC$-content of the word and codes over $F_4$ all of whose words have the same $GC$-content have a particular interest. Therefore our results have some direct applications in this direction.Comment: 16 page