Polynomial root separation in terms of the Remak height

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].

Anahtar Kelimeler:

Polynomial root separation, Mahler's measure, Remak's height, discriminants

Polynomial root separation in terms of the Remak height

Keywords:

Polynomial root separation, Mahler's measure, Remak's height, discriminants,

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