Exceptional units and numbers of small Mahler measure. (English) Zbl 0851.11064

Silverman, Joseph H.

Exp. Math. 4, No. 1, 69-83 (1995).

Let \(\alpha\) be a unit of degree \(d\) in an algebraic number field and assume \(\alpha\) is not a root of unity. Let \(E (\alpha)\) be the number of values of \(n\) for which \(\alpha^n - 1\) is a unit and let \(U (\alpha)\) be the number of values of \(m\) for which \(\Phi_m (\alpha)\) is a unit, where \(\Phi_m (x)\) is the \(m\)th cyclotomic polynomial. The author proves that \(U (\alpha) \leq cd^{1 + 0.7/ \log \log d}\) where \(c\) is an effectively computable constant. This implies a similar estimate for \(E (\alpha)\). The proof is based on an explicit and useful lower bound for \(\Phi_m (\alpha)\). The paper also contains some interesting numerical results for \(\alpha\) of small Mahler measure which lead to the suggestion that perhaps \(U (\alpha) \leq c_1 (d/ \log M (\alpha)) + c_2\), where \(c_1\) and \(c_2\) are absolute constants. Examples of the reviewer are given which show that the first term here cannot be replaced by \(d \psi (d)/ \log M (\alpha)\) for any \(\psi (d) \to 0\) as \(d \to \infty\). Recently, C. Pinner has constructed some interesting examples to show that \(U (\alpha) = \Omega (d^{1/2})\). One simply notes that if \(\alpha\) is a root of the polynomial \(x + \prod_{m \leq N} \Phi_m (x)\), then \(\alpha\) is a unit of degree \(d \ll N^2\) and \(\Phi_m (\alpha)\) is a unit for all \(m \leq N\).

Reviewer: D.W.Boyd (Vancouver)

MSC:

11R27	Units and factorization
11Y40	Algebraic number theory computations
11D61	Exponential Diophantine equations
11R06	PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11J68	Approximation to algebraic numbers

Keywords:

exceptional units; small Mahler measure; cyclotomic polynomial

Software:

PARI/GP

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