In mathematics, a Littlewood polynomial is a polynomial whose coefficients are all either 1 {\displaystyle 1} or − 1 {\displaystyle -1} . Equivalently, its coefficient vector is a finite binary sign sequence. Littlewood polynomials are named after J. E. Littlewood, who studied their values on the unit circle and posed several influential extremal problems about them in the 1960s. The natural size of a Littlewood polynomial of degree n {\displaystyle n} on the unit circle is n + 1 {\displaystyle {\sqrt {n+1}}} , because its normalized L 2 {\displaystyle L^{2}} norm is exactly that value. A central question asked whether the modulus could remain between two fixed positive multiples of n + 1 {\displaystyle {\sqrt {n+1}}} at every point of the circle. Littlewood conjectured that this was possible; the conjecture was proved in 2020 by Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba. Stronger questions about asymptotically constant modulus, extremal L q {\displaystyle L^{q}} norms, Mahler measure, zero distribution, and autocorrelation remain active.
Definition and elementary properties A polynomial
P ( z ) = ∑ j = 0 n a j z j {\displaystyle P(z)=\sum _{j=0}^{n}a_{j}z^{j}}
is a Littlewood polynomial of degree n {\displaystyle n} if a j ∈ { − 1 , 1 } {\displaystyle a_{j}\in \{-1,1\}} for every j {\displaystyle j} . There are 2 n + 1 {\displaystyle 2^{n+1}} such polynomials. Multiplication by − 1 {\displaystyle -1} , the substitution z ↦ − z {\displaystyle z\mapsto -z} , and passage to the reciprocal polynomial
P ∗ ( z ) = z n P ( 1 / z ) {\displaystyle P^{*}(z)=z^{n}P(1/z)}
preserve the class. These operations account for many of the natural symmetries used in enumerations. Every zero α {\displaystyle \alpha } of a Littlewood polynomial satisfies
1 2 < | α | < 2. {\displaystyle {\frac {1}{2}}<|\alpha |<2.}
Indeed, when | z | ≤ 1 / 2 {\displaystyle |z|\leq 1/2} , the constant term has greater modulus than the sum of all remaining terms, and the upper bound follows by applying the same argument to the reciprocal polynomial. For 0 < q < ∞ {\displaystyle 0<q<\infty } , its normalized norm on the unit circle is
‖ P ‖ q = ( 1 2 π ∫ 0 2 π | P ( e i t ) | q d t ) 1 / q , {\displaystyle \lVert P\rVert _{q}=\left({\frac {1}{2\pi }}\int _{0}^{2\pi }|P(e^{it})|^{q}\,dt\right)^{1/q},}
and ‖ P ‖ ∞ = max | z | = 1 | P ( z ) | {\displaystyle \lVert P\rVert _{\infty }=\max _{|z|=1}|P(z)|} . Parseval's identity gives
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