challenges / flat-littlewood-polynomials
Flat Littlewood polynomials, extremes of |p| on the unit circle
AlphaEvolve problem 28: polynomials with +/-1 coefficients whose modulus on the unit circle is as flat as possible: minimise the max (C+), maximise the min (C-), or minimise the annulus width max - min (Cw), each divided by sqrt(n+1). Extremes are certified by branch and bound; scored against Odlyzko's exhaustive-search optima for n = 10, 12, 24 and his skew-symmetric best for n = 102.
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# Flat Littlewood polynomials: keep |p| close to sqrt(n+1) on the whole circle
## Problem
For `n >= 1` let `U_n` be the polynomials `p(z) = c_0 + c_1 z + ... + c_n z^n` with every
`c_k in {+1, -1}`. Parseval gives `mean |p|^2 = n + 1` on the unit circle, so `|p| / sqrt(n+1)`
is the natural normalisation and a perfectly flat polynomial would have it identically 1.
Littlewood asked (1966) whether such "ultraflat" polynomials exist in `U_n`; the extensive
computations of Odlyzko (2018) say almost certainly not, with the extreme values converging to
about `M = 1.27`, `m = 0.64`, `W = 0.79`. Define, over `|z| = 1`,
M(p) = max |p(z)| / sqrt(n+1), m(p) = min |p(z)| / sqrt(n+1), W(p) = M(p) - m(p),
and the degree-`n` optima `C+(n) = min_p M(p)`, `C-(n) = max_p m(p)`, `Cw(n) = min_p W(p)`. The
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