Turkish Journal of Mathematics
Article Title
DOI
10.3906/mat-1201-41
Abstract
We investigate some monic integer irreducible polynomials which have two close roots. If P(x) is a separable polynomial in Z[x] of degree d \geq 2 with the Remak height R(P) and the minimal distance between the quotient of two distinct roots and unity Sep(P), then the inequality 1/Sep(P) \ll R(P)^{d-1} is true with the implied constant depending on d only. Using a recent construction of Bugeaud and Dujella we show that for each d \geq 3 there exists an irreducible monic polynomial P \in Z[x] of degree d for which R(P)^{(2d-3)(d-1)/(3d-5)} \ll 1/Sep(P). For d=3 the exponent 3/2 is improved to 5/3 and it is shown that the exponent 2 is optimal in the class of cubic (not necessarily monic) irreducible polynomials in Z[x].
Keywords
Polynomial root separation, Mahler's measure, Remak's height, discriminants
First Page
747
Last Page
761
Recommended Citation
DUBICKAS, ARTURAS
(2013)
"Polynomial root separation in terms of the Remak height,"
Turkish Journal of Mathematics: Vol. 37:
No.
5, Article 3.
https://doi.org/10.3906/mat-1201-41
Available at:
https://journals.tubitak.gov.tr/math/vol37/iss5/3
