Georges Rhin

Approximants de Padé et mesures effectives d’irrationalité

Les approximants de Pade; des fonctions hypergeometriques ont ete utilises pour l’etude en des points rationnels z=p/q des approximations diophantiennes des valeurs de ces fonctions. Cette methode puissante a donne des resultats rappeles au paragraphe 3. Nous montrerons au paragraphe 4 comment les approximants de Pade peuvent etre remplaces par d’autres polynomes qui fournissent de meilleures mesures d’irrationalite, ameliorant certains resultats de G.V. Chudnovsky. Les definitions necessaires sont rappelees au paragraphe 2.

Integer transfinite diameter and polynomials with small mahler measure

In this work, we show how suitable generalizations of the integer transfinite diameter of some compact sets in C give very good bounds for coefficients of polynomials with small Mahler measure. By this way, we give the list of all monic irreducible primitive polynomials of Z[X] of degree at most 36 with Mahler measure less than 1. 324... and of degree 38 and 40 with Mahler measure less than 1. 31.

Minimal Mahler Measures

We determine the minimal Mahler measure of a primitive irreducible noncyclotomic polynomial with integer coefficients and fixed degree D, for each even degree D ≤ 54. We also compute all primitive irreducible noncyclotomic polynomials with measure less than 1.3 and degree at most 44.

On the absolute Mahler measure of polynomials having all zeros in a sector

Let α be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates α i are confined to a sector |arg z| ≤ θ. We compute the greatest lower bound c(θ) of the absolute Mahler measure (Π i=1 d max(1, |α i |)) 1/d of α, for θ belonging to nine subintervals of [0, 2π/3]. In particular, we show that c(π/2) = 1.12933793, from which it follows that any integer α ¬= 1 and α ¬= e ±iπ/3 all of whose conjugates have positive real part has absolute Mahler measure at least c(π/2). This value is achieved for α satisfying α + 1/α = β 0 2 , where β 0 = 1.3247... is the smallest Pisot number (the real root of β 0 3 = β 0 + 1)

On the smallest value of the maximal modulus of an algebraic integer

The house of an algebraic integer of degree d is the largest modulus of its conjugates. For d 1 of degree d, say m(d). As a consequence we improve Matveevs theorem on the lower bound of m(d). We show that, in this range, the conjecture of SchinzelZassenhaus is satisfied. The minimal polynomial of any algebraic integer α whose house is equal to m(d) is a factor of a bi-, tri- or quadrinomial. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C. They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer α whose house is small.

On the absolute Mahler measure of polynomials having all zeros in a sector. II

kkLet α be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates α i are confined to a sector |arg z| < θ. In the paper On the absolute Mahler measure of polynomials having all zeros in a sector, G. Rhin and C. Smyth compute the greatest lower bound c(θ) of the absolute Mahler measure (Π d i=1 max(1, |α i |)) 1/d of α, for θ belonging to nine subintervals of [0,2π/3]. In this paper, we improve the result to thirteen subintervals of [0,π] and extend some existing subintervals.

New methods providing high degree polynomials with small Mahler measure

In this work, we propose two new methods devoted to provide a large list of new polynomials with high degree and small Mahler measure. First, by statistical considerations, we augment Mossinghoffs list of polynomials with degree at most 180, and then we give a new list of such polynomials of degree up to 300. The second idea is to perturb polynomials of Mossinghoffs list, and for higher degrees, of this new list, and to use them as initial polynomials for a minimization method, which converges to new polynomials with lower Mahler measure.

Two families of periodic Jacobi Algorithms with period lengths going to infinity

We exhibit two infinite families of periodic Jacobi Algorithm expansions with period length equal to 4m+1 in the first case and to 3m+1 in the second case.

Algebraic integers whose conjugates all lie in an ellipse

We find all 15909 algebraic integers α whose conjugates all lie in an ellipse with two of them nonreal, while the others lie in the real interval [-1,2]. This problem has applications to finding certain subgroups of SL(2, C). We use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of C. This gives good bounds for the coefficients of the minimal polynomial of a.

Finding Degree-16 Monic Irreducible Integer Polynomials of Minimal Trace by Optimization Methods

In this paper, we give a list of monic integer polynomials of smallest possible trace, irreducible, of degree 16, and having all roots real and positive. Moreover, we think that this list may be complete.