A curious conjecture about self-reciprocal polynomials

Let p be a polynomial
p(z) = a_0 + a_1z+\dots + a_nz^n\ \ \ \ \ \ a_i\in {\bf C},
and let p^* denote the reciprocal polynomial
p^*(z) = a_n + a_{n-1}z+\dots + a_0z^n=z^np(1/z).
We say p is self-reciprocal if p=p^*. For example,
1+2z+3z^2+2z^3+z^4
and
1+2z+3z^2+3z^3+2z^4+z^5
are self-reciprocal. Suppose
p(z) = a_0 + a_1z+\dots + a_dz^d+a_{d+1}z^{d+1}+a_dz^{d}+\dots +a_{1}z^{2d+1}+a_0z^{2d+2},\ \ \ \ \ \ a_i\in {\bf C},
or
p(z) = a_0 + a_1z+\dots + a_dz^d+a_{d+1}z^{d+1}+a_{d+1}z^{d+2}+a_dz^{d+3}+\dots +a_{1}z^{2d+2}+a_0z^{2d+3},\ \ \ \ \ \ a_i\in {\bf C},

The question is this: for which increasing sequences a_0<a_1<\dots a_{d+1} do the polynomial roots p(z)=0 lie on the unit circle |z|=1?

Some examples:

symmetric-increasing-coeff-plot1
This represents, when d=1 and a_0=1 and a_1=1.1 the largest a_{d+1} which has this roots property is as in the plot.

symmetric-increasing-coeff-plot2
This represents, when d=1 and a_0=1 and a_1=1.2 the largest a_{d+1} which has this roots property is as in the plot.

symmetric-increasing-coeff-plot3
This represents, when d=1 and a_0=1 and a_1=1.3 the largest a_{d+1} which has this roots property is as in the plot.

symmetric-increasing-coeff-plot4
This represents, when d=1 and a_0=1 and a_1=1.4 the largest a_{d+1} which has this roots property is as in the plot.

symmetric-increasing-coeff-plot5
This represents, when d=1 and a_0=1 and a_1=1.5 the largest a_{d+1} which has this roots property is as in the plot.

Conjecture 1: Let s:{\bf Z}_{>0}\to {\bf R}_{>0} be a ”slowly increasing” function.

  1. Odd degree case.
    If g(z)= a_0 + a_1z + \dots + a_dz^d, where a_i=s(i), then the roots of p(z)=g(z)+z^{d+1}g^*(z) all lie on the unit circle.
  2. Even degree case.
    The roots of
    p(z)=a_0 + a_1z + \dots + a_{d-1}z^{d-1} + a_{d}z^{d} + a_{d-1}z^{d+1} + \dots + a_{1}z^{2d-1}+a_0z^{2d}
    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
p(z)=a_0 + a_1z + \dots + a_{d-1}z^{d-1} + a_{d}z^{d} + a_{d-1}z^{d+1} + \dots + a_{1}z^{2d-1}+a_0z^{2d}  ,
with a_0=1 and all a_i>0, is symmetric increasing if and only if it can be written as a product of quadratics of the form x^2-2\cos(\theta)x+1, where all but one of the factors satisfy 2\pi/4\leq \theta\leq 4\pi/3. One of the factors can be of the form x^2+\alpha x+1, for some \alpha \geq 0.

* It was pointed out to me by Els Withers that this conjecture is false.

2 thoughts on “A curious conjecture about self-reciprocal polynomials

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