An inequality about irreducible factors of integer polynomials. II. (English) Zbl 0739.11010

Glesser, Philippe; Mignotte, Maurice

Applied algebra, algebraic algorithms and error-correcting codes, Proc. 8th Int. Conf., AAECC-8, Tokyo/Jap. 1990, Lect. Notes Comput. Sci. 508, 260-266 (1991).

[For the entire collection see Zbl 0727.00017; for part I, cf. J. Number Theory 30, 156-166 (1988: Zbl 0648.12002).]
Let \(F(x)\) be a polynomial with integral coefficients. The height of \(F\), written \(H(F)\), is defined as the maximum of the moduli of the coefficients of \(F\). The authors prove the following useful upper bound for the height of an irreducible factor \(P(x)\) of \(F(x)\), \[ H(P)\leq(e/2)^{\sqrt d}(d+2\sqrt{d}+1)^{1/2+\sqrt d}M(F)^{1+\sqrt d}, \] where \(d=\deg(P)\) and \(M(F)\) denotes the Mahler measure of \(F\).

Reviewer: K.S.Williams (Ottawa)

MSC:

11C08	Polynomials in number theory
12D05	Polynomials in real and complex fields: factorization

Keywords:

integer polynomial; upper bound; height; irreducible factor; Mahler measure

Citations:

Zbl 0727.00017; Zbl 0648.12002