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

✍ Scribed by David W. Boyd

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 436 KB
• Volume: 16
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.1993.1037

✦ Synopsis

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