AVERAGE MAHLER’S MEASURE AND Lp NORMS OF UNIMODULAR POLYNOMIALS

A polynomial f ∈ C[z] is unimodular if all its coefficients have unit modulus. Let Un denote the set of unimodular polynomials of degree n−1, and let Un denote the subset of reciprocal unimodular polynomials, which have the property that f(z) = ωzn−1f(1/z) for some complex number ω with |ω| = 1. We study the geometric and arithmetic mean values of both the normalized Mahler’s measure M(f)/ √ n and Lp norm ‖f‖p / √ n over the sets Un and Un, and compute asymptotic values in each case. We show for example that both the geometric and arithmetic mean of the normalized Mahler’s measure approach e−γ/2 = 0.749306 . . . as n → ∞ for unimodular polynomials, and e−γ/2/ √ 2 = 0.529839 . . . for reciprocal unimodular polynomials. We also show that for large n almost all polynomials in these sets have normalized Mahler’s measure or Lp norm very close to the respective limiting mean value.