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- Georges Rhin
- 2001

1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. .. and µ(ζ(3)) < 13.41782. .. Several improvements on such irrationality measures were subsequently given, and we refer to the introductions of the papers [3] and… (More)

Let a be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates a, are confined to a sector |argz| < 6. We compute the greatest lower bound c(6) of the absolute Mahler measure (n?=i max(l, |a,|))'/d of a , for 6 belonging to nine subintervalsof [0, 2^/3]. In particular, we show that c(7t/2) = 1.12933793, from which it follows… (More)

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… (More)

The house of an algebraic integer of degree d is the largest mod-ulus of its conjugates. For d ≤ 28, we compute the smallest house > 1 of degree d, say m(d). As a consequence we improve Matveev's theorem on the lower bound of m(d). We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic… (More)

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.

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. 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 (d i=1 max(1, |α i |)) 1/d of α, for θ… (More)

We find all 15909 algebraic integers « 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… (More)