2016/07/07 by Charles L. Samuels · 2 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic number #Conjecture #Connection (principal bundle) #Fibonacci number #Mathematical Dynamics and Fractals #Measure (data warehouse) #Polynomial #Sequence (biology) #Series (stratigraphy) #math.NT #msc:11B39 #msc:11G50 #msc:11R04 #msc:11R09 #semigroups and automata theory
paper · pdf · doi:10.1007/s10998-017-0189-9
published in Periodica Mathematica Hungarica 75(2), 221-243 (Springer Science+Business Media)
arxiv created 2016/07/07 · openalex created_date 2016/08/23 · openalex publication_date 2017/07/24 · arxiv updated 2019/12/23 · openalex updated_date 2026/08/05
If α is a non-zero algebraic number, we let m(α) denote the Mahler measure of the minimal polynomial of α over \mathbb Z. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the t-metric Mahler measure, denoted mt(α). For fixed α∈ \mathbb Q, the map t↦ mt(α) is continuous, and moreover, is infinitely differentiable at all but finitely many points, called \it exceptional points for α. It remains open to determine whether there is a sequence of elements αn∈ \mathbb Q such that the number of exceptional points for αn tends to ∞ as n→ ∞. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the t-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.