Exceptional Units and Numbers of Small Mahler Measure
Exceptional Units and Numbers of Small Mahler Measure
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DOI：
10.1080/10586458.1995.10504309
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
J. Silverman
中科院分区：
文献类型：
--
作者：
J. Silverman
设α是代数数域中的d次单位，并假定a不是单位根。我们进行了数值研究，证明了当α具有小Mahler测度时，有许多n值1 - α n为单位，也有许多m值Φ m（α）为单位，其中Φ m为m次分圆多项式.我们证明了Ω m的这种值的个数在O（dl +0.7/log log d）以上有界，并且我们描述了Boyd的一个构造，它给出了Ω（d 0.6/log log d）的一个下界.
Let α be a unit of degree d in an algebraic number field, and assume that a is not a root of unity. We conduct a numerical investigation that suggeststhat if α has small Mahler measure, there are many values of n for which 1 – α n is a unit and also many values of m for which Φ m (α) is a unit, where Φ m is the m-th cyclotomic polynomial. We prove that the number of such values of nand m is bounded above by O(d l+0.7/log log d ), and we describe a construction of Boyd that givesa lower bound of Ω(d 0.6/log log d ).