---
tag: 5A16A9W335
kind: concept
name: Littlewood polynomial
msc: [42A05, 30C10, 26D05]
sources:
  - url: https://en.wikipedia.org/wiki/Littlewood_polynomial
same_as:
  wikidata: Q3395696
  wikipedia: Littlewood polynomial
---

## Summary

A Littlewood polynomial is a polynomial all of whose coefficients are $+1$ or $-1$.
On the unit circle such a polynomial of degree $n$ has mean square $n + 1$, so its typical size there is $\sqrt{n+1}$.
Littlewood asked how close to constant its absolute value on the circle can be.
This question about flat polynomials links harmonic analysis, combinatorics and signal design.
The [[Rudin–Shapiro polynomials|PSAX8W3ZYV]] are a classical family that is flat from above.

## Explanations

### L1 human

Write a polynomial using only the coefficients $+1$ and $-1$, for example $1 + z - z^2$.
On the unit circle $\abs{z} = 1$, the sum of squares of the coefficients controls the average of $\abs{p(z)}^2$.
The question is whether the signs can be chosen so that $\abs{p(z)}$ is nearly the same everywhere on the circle.
Such flat sign sequences are useful in signal processing, where they spread energy evenly over all frequencies.