2025/03/28 by Matilde Laĺın, Lalín, Matilde N., Siva Sankar Nair +5 · 2 citations
Mathematics · #11R06 (Primary) 11M06 #11R42 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2503.22536
openalex publication_date 2025/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We study the areal Mahler measure of the two-variable, k-parameter family x+y+k and prove explicit formulas that demonstrate its relation to the standard Mahler measure of these polynomials. The proofs involve interpreting the areal Mahler measure as a random walk in the complex plane and utilizing the areal analogue of the Zeta Mahler function to arrive at the result. Using similar techniques, we also present formulas for a three-variable family (x+1)(y+1)+kz in terms of the standard Mahler measure, along with terms that involve certain hypergeometric functions. For both families we show that its areal Mahler measure is, up to elementary functions, a linear combination of the normal Mahler measure and the volume of the Deninger cycle of the corresponding family.