Bounds for the Height of a Factor of a Polynomial in Terms of Bombieri's Norms: II. The Smallest Factor

AbstractLet ƒ(x) be a polynomial of degree n with complex coefficients, which factors as ƒ(x) = g(x)h(x). Let H (g) be the maximum of the absolute value of the coefficients of g . For 1 ≤ p ≤ ∞, let [ƒ]p denote the pth Bombieri norm of ƒ. This norm is a weighted ℓp norm of the coefficient vector of ƒ, the weights being certain negative powers of the binomial coefficients. We determine explicit constants D(p) such that H (g)H(h) ≤ D (p)n[ƒ]p which implies that min(H(g), H(h)) ≤ D (p)n/2[ƒ]1/2p. The constants D(p) are proved to be best possible for infinitely many values of p including p = 1,2 and ∞. If ƒ,g and h have real coefficients, and if ƒ2(x) = (-1)n ƒ(x)ƒ(-x), we give explicit constants E(p) so that H (g)H(h) ≤ E (p)n[ƒ2]1/2p. For p = ∞, this gives an easily computed estimate which is better than the classical inequality H(g)H(h) ≤ 2nM(ƒ), where M(ƒ) denotes Mahler's measure of ƒ, a quantity which is more difficult to compute