Jacobi's four-square theorem: Difference between revisions

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{{for|the orthogonal polynomials in several variables|Hall–Littlewood polynomials}}
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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.
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They are named for [[J. E. Littlewood]] who studied them in the 1950s.
 
==Definition==
A polynomial
 
:<math> p(x) = \sum_{i=0}^n a_i x^i \, </math>
 
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
 
:<math>c_1 \sqrt{n+1} \le | p_n(z) | \le c_2 \sqrt{n+1} . \, </math>
 
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}}

Revision as of 08:35, 18 February 2014

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