Let be a polynomial

and let denote the *reciprocal polynomial*

We say is *self-reciprocal* if . For example,

and

are self-reciprocal. Suppose

or

The question is this: for which increasing sequences do the polynomial roots lie on the unit circle ?

Some examples:

This represents, when and and the largest which has this roots property is as in the plot.

This represents, when and and the largest which has this roots property is as in the plot.

This represents, when and and the largest which has this roots property is as in the plot.

This represents, when and and the largest which has this roots property is as in the plot.

This represents, when and and the largest which has this roots property is as in the plot.

**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.

I don’t see the conjecture, but this paper seems quite relevant:

http://imamat.oxfordjournals.org/content/8/3/397.short

Thanks for the reference! I added a vaguely-worded conjecture, the best I have at this point.