For given $n$, consider a polynomial $\sum_{k=0}^na_kz^k$ with all coefficients $a_k\in\{\pm1\}$. I am interested in the following:

How big can the modulus of a non-real root of such a polynomial be?

Wlog we can assume $a_n=1,a_{n-1}=-1$. Then a systematic search for small $n$, looking each time at the 10 or so extremal polynomials, exhibits a clear pattern for the highest coefficients: they come in groups of $3$ or $4$ consecutive +1's or -1's, starting with $z^n-z^{n-1}+++----+++---++++\cdots$. More precisely, the group sizes are displayed here (first line +1's, second line -1's)

```
1 3 3 4 3 3 3 4 3 3 3 4 3 3 4 4 3 3 4 4 3 3 4 4 3 3 4 . . .
1 4 3 3 4 4 3 3 4 4 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 . . .
```

Note that there are never two adjacent groups of size 4 or three of size 3, so the pattern is rather regular, as may be expected.

As $n$ grows, the extremal root $z_0$ (and $\bar z_0$) of the extremal polynomials does not at all jump around, but converges quite rapidly towards $0.93757749648487973269811306454355 \pm 1.2634174429011374851417570421775\; i$, e.g. we have

$n=20\implies z_0\approx 0.937537 \pm 1.26337\;i$

$n=40\implies z_0\approx 0.9375774916 \pm 1.263417437\;i$

$n=60\implies z_0\approx 0.9375774964839 \pm 1.26341744290078\;i$.

$n=80\implies z_0\approx 0.93757749648487963 \pm 1.263417442901137422\;i$.

$n=100\implies z_0\approx 0.937577496484879732688 \pm 1.263417442901137485132\;i$.

The obvious questions:

Can the above 3-4-sequences be characterized? E.g. are they periodic? or self-similar?What can be said about the limit value of $z_0$? Is there a (closed form or whatever) formula for $z_0$?

Although the problem is not directly related to *Garsia numbers* (see the article mentioned here), it naturally leads again to the question of limit points of zeros of such polynomials, given that $z_0$ visibly yields one of those:

Are there only countably many non-real limit points of zeros of $\pm1$-polynomials?

minimummodulus root $z_n$ is clearly equivalent, since we can consider reciprocal polynomials. However, the minimum modulus problem seems more suitable to pass to the limit, since limits of $\pm$-polynomials are $\pm$-series, with radius of convergence 1, thus including the (any) limit of $z_n$. So one may start from the minimum root problem for $\pm$-series. $\endgroup$ – Pietro Majer Nov 9 '14 at 21:58