Two new points in the spectrum of the absolute Mahler measure of totally positive algebraic integers

نویسنده

  • Valérie Flammang
چکیده

For totally positive algebraic integers α 6= 0, 1 of degree d(α), we consider the set L of values of M(α) 1 d(α) = Ω(α), where M(α) is the Mahler measure of α. C. J. Smyth has found the four smallest values of L and conjectured that the fifth point is Ω((2 cos 2π 60 )2). We prove that this is so and, moreover, we give the sixth point of L.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Computing totally positive algebraic integers of small trace

We construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials. Many new examples of such polynomials of minimal absolute trace (for given degree) are found. The computations are pushed to degrees that previously were unattainable, and one consequence is that the new examples form the majority of ...

متن کامل

On a Problem of Mahler Concerning the Approximation of Exponentials and Logarithms by Michel WALDSCHMIDT Bon

We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic. §1. Two Conjectures on Diophantine Approximation of Logarithms of Algebraic Numbers In 1953 K. Mahler [7] proved that for any sufficiently large positive integers a and b, the estimates log a ≥ a −40 l...

متن کامل

On Normal Numbers and Powers of Algebraic Numbers

Let α > 1 be an algebraic number and ξ > 0. Denote the fractional parts of ξαn by {ξαn}. In this paper, we estimate a lower bound for the number λN (α, ξ) of integers n with 0 ≤ n < N and {ξα} ≥ min { 1 L+(α) , 1 L−(α) } . Our results show, for example, the following: Let α be an algebraic integer with Mahler measure M(α) and ξ > 0 an algebraic number with ξ #∈ Q(α). Put [Q(α, ξ) : Q(α)] = D. T...

متن کامل

Salem Numbers of Trace -2 and Traces of Totally Positive Algebraic Integers

Until recently, no Salem numbers were known of trace below −1. In this paper we provide several examples of trace −2, including an explicit infinite family. We establish that the minimal degree for a Salem number of trace −2 is 20, and exhibit all Salem numbers of degree 20 and trace −2. Indeed there are just two examples. We also settle the closely-related question of the minimal degree d of a...

متن کامل

Counting Perron Numbers by Absolute Value

We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston [Thu].

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Math. Comput.

دوره 65  شماره 

صفحات  -

تاریخ انتشار 1996