Bounds for the Height of a Factor of a Polynomial in Terms of Bombieriʹs Norms: I. The Largest Factor

Let ƒ(x) be a monic polynomial of degree n with complex coefficients, which factors as ƒ(x) = g (x)h(x), where g and h are monic. Let H(g) be the maximum of the absolute value of the coefficients of g. For 1 ≤ p ≤ ∞, let [ƒ]p denote the pthBombieri 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 C(p) such that H(g) ≤ C (p)n[ƒ]p. For p = 2 our result improves a result of Beauzamy. The constants C(1) = 1.38135 … andC (2) = (1 + )/2 = 1.61803 … are proved to be best possible. It is conjectured that C(∞) = 2.17601 … is also best possible, and it is shown that the best constant in this case can be no smaller than 2.14587 …. It is expected that these results will have some application to algorithms for proving the irreducibility of polynomials over the integers.