- Turkish Journal of Mathematics
- Vol: 37 Issue: 5
- Polynomial root separation in terms of the Remak height
Polynomial root separation in terms of the Remak height
Authors : Arturas Dubickas
Pages : 747-761
Doi:10.3906/mat-1201-41
View : 14 | Download : 5
Publication Date : 9999-12-31
Article Type : Makaleler
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