Heights of polynomials over lemniscates

We consider a family of heights defined by the $L_p$ norms polynomials with respect to equilibrium measure lemniscate for $0 \le p \infty$, where $p=0$ corresponds geometric mean (the generalized Mahler measure) and $p=\infty$ standard supremum norm. This special choice allows find an explicit form polynomial, estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish lowest height, also show their uniqueness. discuss relations between results on analogues that include generalizations Kronecker's theorem algebraic integers in unit disk, as well Lehmer's conjecture.