2017/05/22 by Charles L. Samuels
Mathematics · #Algebraic Geometry and Number Theory #Algebraic number #Algebraic number field #Alpha (finance) #Analytic Number Theory Research #Class number #Combinatorics #Computer science #Discrete mathematics #Field (mathematics) #Geometry #Infimum and supremum #Mathematical analysis #Mathematics #Measure (data warehouse) #Meromorphic and Entire Functions #Metric (unit) #Pure mathematics #Quadratic equation #Quadratic field #Quadratic function #Rational number #Real number #Statistics #math.NT #msc:11G50 #msc:11R04 #msc:11R11 #msc:11R27 #msc:11R29 #msc:11R37 #msc:13A15
paper · pdf · doi:10.1007/s10474-017-0770-y
published in Acta Mathematica Academiae Scientiarum Hungaricae 154(1), 105-123 (Springer Nature) · 12 pages
arxiv created 2017/05/22 · openalex publication_date 2017/10/19 · arxiv updated 2019/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For an algebraic number α, the metric Mahler measure m1(α) was first studied by Dubickas and Smyth in 2001 and was later generalized to the t-metric Mahler measure mt(α) by the author in 2010. The definition of mt(α) involves taking an infimum over a certain collection N-tuples of points in \mathbb Q, and from previous work of Jankauskas and the author, the infimum in the definition of mt(α) is attained by rational points when α∈ \mathbb Q. As a consequence of our main theorem in this article, we obtain an analog of this result when \mathbb Q is replaced with any imaginary quadratic number field of class number equal to 1. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases.