Dergiler / Turkish Journal of Mathematics / 2013 / Cilt: 37 - Sayı: 5
Polynomial root separation in terms of the Remak height
- Sayfa
- 747–761
- DOI
- —
Özet
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].
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].