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Heights of polynomials over lemniscates

Volume 198 / 2021

Igor Pritsker Acta Arithmetica 198 (2021), 219-231 MSC: Primary 11R06, 12D10; Secondary 30C10, 30C15. DOI: 10.4064/aa200109-9-11 Published online: 8 February 2021


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 us to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish explicit polynomials of lowest height, and also show their uniqueness. We discuss relations between the standard results on the Mahler measure and their analogues for lemniscates that include generalizations of Kronecker’s theorem on algebraic integers in the unit disk, as well as of Lehmer’s conjecture.


  • Igor PritskerDepartment of Mathematics
    Oklahoma State University
    Stillwater, OK 74078, U.S.A.

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