Let be a polynomial
and let denote the reciprocal polynomial
We say is self-reciprocal if . For example,
are self-reciprocal. Suppose
The question is this: for which increasing sequences do the polynomial roots lie on the unit circle ?
Conjecture 1: Let be a ”slowly increasing” function.
- Odd degree case.
If , where , then the roots of all lie on the unit circle.
- Even degree case.
The roots of
all lie on the unit circle.
The next conjecture gives an idea as to how fast a “slowly increasing” function can grow.
Conjecture 2:* Consider the even degree case only. The polynomial
with and all , is symmetric increasing if and only if it can be written as a product of quadratics of the form , where all but one of the factors satisfy . One of the factors can be of the form , for some .
* It was pointed out to me by Els Withers that this conjecture is false.