Root separation for irreducible integer polynomials 1 Introduction

Root separation for irreducible integer polynomials Yann Bugeaud and Andrej Dujella

Abstract We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and G¨otze (for odd degree).

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Introduction

The height H(P ) of an integer polynomial P (x) is the maximum of the absolute values of its coefficients. For a separable integer polynomial P (x) of degree d ≥ 2 and with distinct roots α1 , . . . , αd , we set sep(P ) = and define e(P ) by

min

1≤i 0 and every sufficiently large a, we have Pd,a (x0 ± (δ0 + ε)a−d and

2 +d/2+1

Pd,a (x0 ± (δ0 − ε)a−d

2 +d/2+1

)>0

) < 0.

This shows that Pd,a (x) has a root x1 in the interval (

x0 − (δ0 + ε)a−d

2 +d/2+1

, x0 − (δ0 − ε)a−d

2 +d/2+1

)

and a root x2 in the interval ( ) 2 2 x0 + (δ0 − ε)a−d +d/2+1 , x0 + (δ0 + ε)a−d +d/2+1 . This yields

sep(Pd,a ) ≤ 2(δ0 + ε)a−d

2 +d/2+1

.

Since H(Pd,a ) = O(a2d−2 ), this gives eirr (d) ≥

2d2 − d − 2 d d−2 = + , 4(d − 1) 2 4(d − 1)

by fixing the arbitrarily small positive real number ε and letting a tend to infinity. The proof of Theorem 1 is complete.

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[5] Y. Bugeaud, Approximation by algebraic numbers. Cambridge Tracts in Mathematics, Cambridge, 2004. [6] Y. Bugeaud, Mahler’s classification of numbers compared with Koksma’s, II. In: Diophantine approximation, 107–121, Dev. Math., 16, Springer Wien New York, Vienna, 2008. [7] Y. Bugeaud and M. Mignotte, On the distance between roots of integer polynomials, Proc. Edinburgh Math. Soc. 47 (2004), 553–556. [8] Y. Bugeaud and M. Mignotte, Polynomial root separation, Intern. J. Number Theory 6 (2010), 587–602. [9] J.-H. Evertse, Distances between the conjugates of an algebraic number, Publ. Math. Debrecen 65 (2004), 323–340. [10] K. Mahler, An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257–262. [11] A. Sch¨onhage, Polynomial root separation examples, J. Symbolic Comput. 41 (2006), 1080–1090.

Yann Bugeaud Universit´e de Strasbourg D´epartement de Math´ematiques 7, rue Ren´e Descartes 67084 Strasbourg, France E-mail address: [email protected] Andrej Dujella Department of Mathematics University of Zagreb Bijeniˇcka cesta 30 10000 Zagreb, Croatia E-mail address: [email protected]

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