Abstract: We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and $p=\infty$ corresponds to the standard supremum norm. This special choice of the measure allows to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. As a consequence, we establish explicit polynomials of minimal height, and also show their uniqueness. We plan to conclude with a discussion of standard results on the regular Mahler measure, and the corresponding analogues on lemniscates.
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