Lehmer's problem for polynomials with odd coefficients

• 成果类型: Article
• 署名作者: Borwein, Peter; Dobrowolski, Edward; Mossinghoff, Michael J.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2007.166.347
• 发表日期: 2007
• 页码: 347-366
• 关键词: height

摘要

We prove that if f(x) = Sigma(n-1)(k=0)a(k)x(k) is a polynomial with no cyclotomic factors whose coefficients satisfy a(k) equivalent to 1 mod 2 for 0 <= k < n, then Mahler's measure of f satisfies log m(f) >= log 5/4 (1-1/n). This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients, degree n - 1, and at least one noncyclotomic factor, then at least one root a of f satisfies vertical bar alpha vertical bar > 1+log 3/2n, resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials, where each coefficient satisfies a(k) equivalent to 1 mod m for a fixed integer m >= 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy a(k) equivalent to I mod p for each k, for a fixed prime p. Last, we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers whose minimal polynomials have coefficients in {- 1, 1}.