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| | {{for|the orthogonal polynomials in several variables|Hall–Littlewood polynomials}} |
| | 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. |
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| | ==Definition== |
| | A polynomial |
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| | :<math> p(x) = \sum_{i=0}^n a_i x^i \, </math> |
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| | is a ''Littlewood polynomial'' if all the <math>a_i = \pm 1</math>. ''Littlewood's problem'' asks for constants ''c''<sub>1</sub> and ''c''<sub>2</sub> such that there are infinitely many Littlewood polynomials ''p''<sub>''n''</sub> , of increasing degree ''n'' satisfying |
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| | :<math>c_1 \sqrt{n+1} \le | p_n(z) | \le c_2 \sqrt{n+1} . \, </math> |
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| | for all <math>z</math> on the unit circle. The [[Rudin-Shapiro polynomials]] provide a sequence satisfying the upper |
| | bound with <math>c_2 = \sqrt 2</math>. |
| | No sequence is known (as of 2008) which satisfies the lower bound. |
| | |
| | ==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 }} |
| | |
| | [[Category:Polynomials]] |
| | [[Category:Conjectures]] |
| | |
| | {{algebra-stub}} |
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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
is a Littlewood polynomial if all the 
. Littlewood's problem asks for constants c1 and c2 such that there are infinitely many Littlewood polynomials pn , of increasing degree n satisfying
for all 
on the unit circle. The Rudin-Shapiro polynomials provide a sequence satisfying the upper
bound with 
.
No sequence is known (as of 2008) which satisfies the lower bound.
References
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