Littlewood polynomial

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{{#invoke:Hatnote|hatnote}} In mathematics, a Littlewood polynomial is a polynomial 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 the complex plane. The answer to this would yield information about the autocorrelation of binary sequences. They are named for J. E. Littlewood who studied them in the 1950s.

Definition

A polynomial

${\displaystyle p(x)=\sum _{i=0}^{n}a_{i}x^{i}\,}$

is a Littlewood polynomial if all the ${\displaystyle a_{i}=\pm 1}$. Littlewood's problem asks for constants c1 and c2 such that there are infinitely many Littlewood polynomials pn , of increasing degree n satisfying

${\displaystyle c_{1}{\sqrt {n+1}}\leq |p_{n}(z)|\leq c_{2}{\sqrt {n+1}}.\,}$

for all ${\displaystyle z}$ on the unit circle. The Rudin-Shapiro polynomials provide a sequence satisfying the upper bound with ${\displaystyle c_{2}={\sqrt {2}}}$. No sequence is known (as of 2008) which satisfies the lower bound.

References

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