Soit $\beta$ un nombre de Pisot ; nous montrons que pour tout entier $n$ assez grand il existe une matrice carrée à coefficients positifs ou nuls dont l’ordre est égal au degré de $\beta$ et dont ${\beta }^n$ est valeur propre.Soit $\beta =a_1/\beta +a_2/{\beta }^2+\cdots +a_n/{\beta }^n+\cdots$ le $\beta$-développement de $\beta$ ; si $\beta$ est un nombre de Pisot, alors la suite ${\left(a_n\right)}_{n\ge 1}$ est périodique après un certain rang $n_0$ (pour $n\ge n_0$, $a_{n+k}=a_n$) et le polynôme$$X^{n_0+k}-(a_1{X}^{n_0+k-1}+\cdots +a_{n_0+k})-(X^{n_0}-(a_1{X}^{n_0}+\cdots +a_{n_0}))$$est appelé polynôme de Parry. Nous montrons qu’il existe un ensemble relativement dense d’entiers $n$ tels que le polynôme minimal de ${\beta }^n$ est égal à son polynôme...

We prove that there exist at least cd⁵ monic irreducible nonreciprocal polynomials with integer coefficients of degree at most d whose Mahler measures are smaller than 2, where c is some absolute positive constant. These polynomials are constructed as nonreciprocal divisors of some Newman hexanomials $1+x^{r₁}+\cdots +x^{r₅}$, where the integers 1 ≤ r₁ < ⋯ < r₅ ≤ d satisfy some restrictions including $2r_j<r_{j+1}$ for j = 1,2,3,4. This result improves the previous lower bound cd³ and seems to be closer to the correct value of...

The main result of this paper implies that for every positive integer $d⩾2$ there are at least ${(d-3)}^2/2$ nonconjugate algebraic numbers which have their Mahler measures lying in the interval $(1,2)$. These algebraic numbers are constructed as roots of certain nonreciprocal quadrinomials.