Roots of Littlewood polynomials with degree 15

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Littlewood polynomial is a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

all of whose coefficients are +1 or −1. Littlewood's problem asks how large the values of such a polynomial must be on the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

. The answer to this would yield information about the

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

of binary sequences. They are named for

J. E. Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to mathematical analysis, analysis, number theory, and differential equations, and had lengthy collaborations with G. H. H ...

who studied them in the 1950s.

Definition

A polynomial :

p(x) = \sum_^n a_i x^i \,

is a ''Littlewood polynomial'' if all the

a_i = \pm 1

. ''Littlewood's problem'' asks for constants ''c''₁ and ''c''₂ such that there are infinitely many Littlewood polynomials ''p''_''n'' , of increasing degree ''n'' satisfying :

c_1 \sqrt \le ,  p_n(z) ,  \le c_2 \sqrt . \,

for all

z

on the unit circle. The Rudin–Shapiro polynomials provide a sequence satisfying the upper bound with

c_2 = \sqrt 2

. In 2019, an infinite family of Littlewood polynomials satisfying both the upper and lower bound was constructed by Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba.

References

* * *{{cite journal , arxiv=1907.09464 , last1=Balister , first1=Paul , last2=Bollobás , first2=Béla , last3=Morris , first3=Robert , last4=Sahasrabudhe , first4=Julian , last5=Tiba , first5=Marius , title=Flat Littlewood Polynomials Exist , year=2019 Polynomials Conjectures