Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials Pm(a) of degree m, defined by Boros and Moll via Pm(a) = 2m+3/2 π (a + 1)m+1/2 ∞ 0 dx (x4 + 2ax2 +1)m+1 , approach the lemniscate {ζ ∈ C: |ζ 2 −1| = 1, ζ <0}, as m→∞.  2004 Elsevier Inc. All rights reserved