The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal
Authors: Erdélyi Tamás 
Published: November 2020

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper K Subscript n Baseline colon-equal left-brace upper Q Subscript n Baseline colon upper Q Subscript n Baseline left-parenthesis z right-parenthesis equals sigma-summation Underscript k equals 0 Overscript n Endscripts a Subscript k Baseline z Superscript k Baseline comma a Subscript k Baseline element-of double-struck upper C comma StartAbsoluteValue a Subscript k Baseline EndAbsoluteValue equals 1 right-brace period">

K

n

≔

{

Q
n

:

Q
n

(
z
)
=

∑

k
=
0

n

a
k

z
k

,

a
k

∈

C

,

|

a
k

|

=
1
}

.

\begin{equation*} {\mathcal {K}}_n \coloneq \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \quad a_k \in {\mathbb {C}}\,, \quad |a_k| = 1 \right \}\,. \end{equation*}

A sequence

(

P
n

)

(P_n)

of polynomials

P
n

∈

K

n

P_n \!\in \! {\mathcal {K}}_n

is called ultraflat if

(
n
+
1

)

−
1

/

2

|

P
n

(

e

i
t

)

|

(n + 1)^{-1/2}|P_n(e^{it})|

converge to

1
1

uniformly in

t

∈

R

t \!\in \! {\mathbb {R}}

. In this paper we prove that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over 2 pi EndFraction integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue left-parenthesis upper P Subscript n Baseline minus upper P Subscript n Superscript asterisk Baseline right-parenthesis left-parenthesis e Superscript i t Baseline right-parenthesis EndAbsoluteValue Superscript q Baseline d t tilde StartStartFraction 2 Superscript q Baseline normal upper Gamma left-parenthesis StartFraction q plus 1 Over 2 EndFraction right-parenthesis OverOver normal upper Gamma left-parenthesis StartFraction q Over 2 EndFraction plus 1 right-parenthesis StartRoot pi EndRoot EndEndFraction n Superscript q slash 2">

1

2
π

∫
0

2
π

|
(

P
n

−

P
n
∗

)
(

e

i
t

)
|

q

d
t

∼

2

q

Γ

(

q
+
1

2

)

Γ

(

q
2

+
1
)

π

n

q

/

2

\begin{equation*} \frac {1}{2\pi } \int _0^{2\pi }{\left | (P_n - P_n^*)(e^{it}) \right |^q \, dt} \sim \frac {{2}^q \Gamma \left (\frac {q+1}{2} \right )}{\Gamma \left (\frac q2 + 1 \right ) \sqrt {\pi }} \,\, n^{q/2} \end{equation*}

for every ultraflat sequence

(

P
n

)

(P_n)

of polynomials

P
n

∈

K

n

P_n \in {\mathcal {K}}_n

and for every

q
∈
(
0
,
∞
)

q \in (0,\infty )

, where

P
n
∗

P_n^*

is the conjugate reciprocal polynomial associated with

P
n

P_n

,

Γ
\Gamma

is the usual gamma function, and the

∼
\sim

symbol means that the ratio of the left and right hand sides converges to

1
1

as

n
→
∞

n \rightarrow \infty

. Another highlight of the paper states that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over 2 pi EndFraction integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue left-parenthesis upper P prime Subscript n Baseline minus upper P Subscript n Superscript asterisk prime Baseline right-parenthesis left-parenthesis e Superscript i t Baseline right-parenthesis EndAbsoluteValue squared d t tilde StartFraction 2 n cubed Over 3 EndFraction">

1

2
π

∫
0

2
π

|
(

P
n
′

−

P
n

∗
′

)
(

e

i
t

)
|

2

d
t

∼

2

n
3

3

\begin{equation*} \frac {1}{2\pi }\int _0^{2\pi }{\left | (P_n^\prime - P_n^{*\prime })(e^{it}) \right |^2 \, dt} \sim \frac {2n^3}{3} \end{equation*}

for every ultraflat sequence

(

P
n

)

(P_n)

of polynomials

P
n

∈

K

n

P_n \in {\mathcal {K}}_n

. We prove a few other new results and reprove some interesting old results as well.