Algebraic Numbers Close to 1 and Variants of Mahler's Measure

✍ Scribed by Francesco Amoroso

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 566 KB
• Volume: 60
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0114

✦ Synopsis

Given a rational function R and a real number p 1, we define h p (R) as the L p norm of max[log |R|, 0] on the unit circle. In this paper we study the behaviour of h p (R) providing various bounds for it. Our results lead to an explicit construction of algebraic numbers close to 1 having small Mahler's measure and small degree, which shows that a lower bound for the distance |:&1| recently given by M Mignotte and M. Waldschmidt is also sharp. From our bounds also follows a statement on polynomials equivalent to the Riemann hypothesis.