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)