Torsten Sillke, 1995

Define a series of polynomials in Z[x] as follows:

  f (x) = 0
   1
           ___ 
           \      i(n-i)
  f (x) =   >    x       (1 - f (x))
   n       /__,                i
        0 < i < n

The roots of f_n have the form of a lollipop.
You get some complex roots near the unit circle
and some real roots >= -1.

>>> Are the roots of this series bounded? <<<


The distribution of the roots is like the one of
random polynomials.

1) The complex roots:
  If the coefficients are complex independent standard
  normals, the zeros concentrate on the unit circle
  (not the disc!) as the degree grows.

2) The real roots:
  The expected number of real zeros E_n of a random
  polynomial of degree n with independent standard normal
  coefficients is:
    E_n = 2/Pi * ln(n) + 0.6257358072 + 2/(Pi*n) + O(1/(n*n))

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References (roots of random polynomials):

Alan Edelman, Eric Kostlan;
  How many zeros of a random polynomial are real?
  Bulletin of the AMS 32:1 (Jan. 1995) 1-37
  (Gives 51 references. The following refs. are from here.)

A. M. Odlyzko, B. Poonen;
  Zeros of polynomials with 0, 1 coefficients,
  Enseign. Math. 39 (1993) 317-348

P. Erd"os, P. Turan;
  On the distribution of roots of polynomials,
  Ann. of Math. (2) 51 (1950) 105-119

A. T. Bharucha-Reid, M. Sambandham;
  Random polynomials,
  Academic Press, New York (1986)