If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in X. In spite of this we are able to construct a symbolic space V and a class of shift-invariant probability measures ...

The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in Magma using the very sophisticated algorithm for computing class groups of algebraic number fields written by Steve Donnelly it is shown that the extremal even unimodular lattice Γ72 from [17] is the unique extremal even unimodular lattice of dimension 72 that a...

A Wedderburn polynomial over a division ring K is a minimal polynomial of an algebraic subset of K. Special cases of such polynomials include, for instance, the minimal polynomials (over the center F = Z(K)) of elements of K that are algebraic over F . In this note, we give a survey on some of our ongoing work on the structure theory of Wedderburn polynomials. Throughout the note, we work in th...

Let f(x) ∈ Z[x] be a totally real polynomial with roots α1 ≤ . . . ≤ αd. The span of f(x) is defined to be αd − α1. Monic irreducible f(x) of span less than 4 are special. In this paper we give a complete classification of those small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. As one application, we find some low-degree polynomials that do not aris...

Abstract. A directed graph G is nonderogatory if its adjacency matrix A is nonderogatory, i.e., the characteristic polynomial of A is equal to the minimal polynomial of A. Given integers r ≥ 2 and h ≥ 3, a directed windmill Mh (r) is a directed graph obtained by coalescing r dicycles of length h in one vertex. In this article we solve a conjecture proposed by Gan and Koo ([3]): Mh (r) is nonder...

We prove that if f(x) = ∑n−1 k=0 akx k is a polynomial with no cyclotomic factors whose coefficients satisfy ak ≡ 1 mod 2 for 0 ≤ k < n, then Mahler’s measure of f satisfies log M(f) ≥ log 5 4 ( 1 − 1 n ) . This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients, degree n−1, and at least one noncyclotomic fact...

Given a number β > 1, the beta-transformation T = Tβ is defined for x ∈ [0, 1] by Tx := βx (mod 1). The number β is said to be a betanumber if the orbit {Tn(1)} is finite, hence eventually periodic. In this case β is the root of a monic polynomial R(x) with integer coefficients called the characteristic polynomial of β. If P (x) is the minimal polynomial of β, then R(x) = P (x)Q(x) for some pol...

Matrix functions are used in many areas of linear algebra and arise in numerical applications in science and engineering. In this paper, we introduce an effective approach for determining matrix function f(A)=g(q(A)) of a square matrix A, where q is a polynomial function from a degree of m and also function g can be a transcendental function. Computing a matrix function f(A) will be time- consu...

Algorithms for computing integral bases of an algebraic function eld are implemented in some computer algebra systems. They are used e.g. for the integration of algebraic functions. The method used by Maple 5.2 and AXIOM is given by Trager in Trager,1984]. He adapted an algorithm of Ford and Zassenhaus Ford,1978], that computes the ring of integers in an algebraic number eld, to the case of a f...