Abstract: The classical Mahler measure may be defined by an integral over the normalized arc length measure. From this definition and an equivalent closed form, many results of the classical Mahler measure are well known and have been extensively studied. We consider two generalizations of the classical Mahler measure, the Mahler measure over lemniscates and the areal Mahler measure over lemniscates. The former measure provides a natural generalization of the classical Mahler measure and is defined by an analogous integral over the equilibrium measure of a lemniscate L. The latter measure is defined similarly, but with integration instead done with respect an areal measure over the filled lemniscate E. For both generalizations, we explore the natural analogues of each to results from the classical Mahler measure. In particular, we focus on analogues of results by Kronecker, Schinzel, and Lehmer. This is a joint work with Igor Pritsker.
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