Quaternionic Mahler measure

Published 5 Mar 2024 in math.NT and math.CV | (2403.02851v1)

Abstract: We introduce the quaternionic Mahler measure for non-commutative polynomials, extending the classical complex Mahler measure. We establish the existence of quaternionic Mahler measure for slice regular polynomials in one and two variables. We study the quaternionic Mahler measure for real and slice regular polynomials, and consider the associated Lehmer problem. Various formulas of quaternionic Mahler measures are proved.

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References (1)

S. Eilenberg, I. Niven, The “fundamental theorem of algebra” for quaternions, Bulletin of the American Mathematical Society (4) 50 (1944), pp. 246–248.

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