Littlewood polynomial

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

:$p(x)\; =\; sum\_\{i=0\}^n\; a\_i\; x^i\; ,$

is a "Littlewood polynomial" if all the $a\_i\; =\; pm\; 1$. Let ||"p"|| denote the supremum of |"p"("z")| on the unit circle. "Littlewood's problem" asks for constants "c"₁ and "c"₂ such that there are infinitely many "p"_"n" , of increasing degree "n", such that

:$c\_1\; sqrt\{n+1\}\; le\; Vert\; p\_n\; Vert\; le\; c\_2\; sqrt\{n+1\}\; .\; ,$

The Rudin-Shapiro polynomials provide a sequence satisfying the upper bound with $c\_2\; =\; sqrt\; 2$.No sequence is known (as of 2008) which satisfies the lower bound.

ee also

* Hall–Littlewood polynomial
* Littlewood's conjecture

References

*cite book | author=Peter Borwein | authorlink=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | series=CMS Books in Mathematics | publisher=Springer-Verlag | year=2002 | isbn=0-387-95444-9 | pages=2-5,121-132
*cite book | author=J.E. Littlewood | authorlink=J. E. Littlewood | title=Some problems in real and complex analysis | publisher=D.C. Heath | year=1968