AGarsia number is an algebraic integer of norm ±2 such that all of the roots of its minimal polynomial are strictly greater than 1 in absolute value. Little is known about the structure of the set of Garsia numbers. The only known limit point of positive real Garsia numbers was 1 (given, for example, by the set of Garsia numbers 21/n). Despite this, there was no known interval of [1,2] where th...

Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we ...

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show...

We present a new conjecture relating the minimal polynomial solution of the level-one U q (sl(2)) quantum Knizhnik-Zamolodchikov equation for generic values of q in the link pattern basis and some q-enumeration of Totally Symmetric Self-Complementary Plane Partitions.

Metric heights are modified height functions on the non-zero algebraic numbers Q ∗ which can be used to define a metric on certain cosets of Q ∗ . They have been defined with a view to eventually applying geometric methods to the study of Q ∗ . In this paper we discuss the construction of metric heights in general. More specifically, we study in some detail the metric height obtained from the n...

In this paper we project to develop two methods for computing the principal matrix pth root. Our approach makes use of the notion of primary matrix functions and minimal polynomial. Therefore, compact formulas for the principal matrix pth root are established and significant cases are explored. Mathematics Subject Classification: Primary 15A24, 15A99, 65H10 Secondary 15A18

Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closed-form”) solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semiinvariants and a theorem in Singer and Ulmer (1997) in such...

We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forestlike partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerat...

We introduce a class of dynamical systems of algebraic origin, consisting of selfinteracting irreducible polynomials over a field. A polynomial f is made to act on a polynomial g by mapping the roots of g. This action identifies a new polynomial h, as the minimal polynomial of the displaced roots. By allowing several polynomials to act on one another, we obtain a self-interacting system with a ...

Matrix theory and its applications make wide use of the eigenprojections of square matrices. The paper demonstrated that the eigenprojection of a matrix A can be calculated with the use of any annihilating polynomial for A, where u ≥ indA. This enables one to establish the components and the minimum polynomial of A, as well as the Drazin inverse A.