Apollonius circles and the number of irreducible factors of polynomials

Apollonius circles and irreducibility criteria for polynomials

We prove the irreducibility of integer polynomials $f(X)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscisae $a$ and $b$, with ratio of the distances to these points depending on the canonical decomposition of $f(a)$ and $f(b)$. In particular, we obtain irreducibility criteria for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is either an Eneström-Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.

Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers

Concentration and equi-distribution, near the unit circle, in Solomyak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number $β$ are characterized by means of the Erdős-Turán approach, and its improvements by Mignotte and Amoroso, applied to the analytical function $f_β(z) = -1 + \sum_{i \geq 1} t_i z^i$ associated with the Rényi $β$-expansion $d_β(1)= 0.t_1 t_2 ...$ of unity. Mignotte's discrepancy function requires the knowledge of the factorization of the Parry polynomial of $β$. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaa…

基 $b(T)$ 下数字受限的不可约多项式

Irreducible polynomials with restricted digits in base $b(T)$

本文研究有限域上多项式环中数字受限的不可约多项式计数，利用傅里叶分析得到渐近公式，并推广圆法至任意多项式基。