**Question P.** *Can a polynomial $P(x)=\sum\_{n=0}^ma\_nx^n$ with coefficients $a\_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?*


Also I am interested in a similar question for analytic functions.


**Question A.** *Let $f(x)=\sum\_{n=0}^\infty a\_nx^n$ a series with coefficients $a\_n\in\{-1,1\}$ such that $\sup\_{m\in\mathbb N}|\sum\_{n=0}^ma\_i|<\infty$. Can the analytic function $f$ have a multiple root in the interval $(0,1)$?*



Multiple roots of polynomials with coefficients $\pm 1$

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> **Question P.** *Can a polynomial $P(x)=\sum\_{n=0}^ma\_nx^n$ with coefficients $a\_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?*
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Yes. The following four Littlewood polynomials:


* $z^{27} + z^{26} + z^{25} + z^{24} + z^{23} - z^{22} - z^{21} + z^{20} + z^{19} + z^{18} - z^{17} - z^{16} - z^{15} - z^{14} - z^{13} - z^{12} - z^{11} - z^{10} + z^9 + z^8 - z^7 + z^6 - z^5 + z^4 - z^3 + z^2 + z - 1 = (z^{18} + z^{16} + 2z^{15} + 2z^{13} + z^{12} + 2z^{11} + 3z^{10} + 3z^8 + 2z^7 + z^6 + 2z^5 + 2z^3 + 1)(z^2 + 1)(z - 1)(z^3 + z^2 - 1)^2$
* $z^{27} + z^{26} + z^{25} - z^{24} - z^{23} - z^{22} + z^{21} - z^{20} - z^{19} + z^{18} + z^{17} - z^{16} - z^{15} + z^{14} + z^{13} - z^{12} - z^{11} - z^{10} - z^9 - z^8 + z^7 + z^6 - z^5 + z^4 - z^3 + z^2 + z - 1 = (z^{21} - z^{20} + 2z^{19} - 2z^{18} + z^{17} + z^{16} - 3z^{15} + 3z^{14} - 2z^{13} + 2z^{11} - 4z^{10} + 4z^9 - 2z^8 - z^7 + 3z^6 - 4z^5 + 2z^4 - z^3 - z^2 + z - 1)(z^3 + z^2 - 1)^2$
* $z^{27} + z^{26} + z^{25} + z^{24} + z^{23} - z^{22} - z^{21} + z^{20} - z^{19} - z^{18} - z^{17} + z^{16} - z^{15} + z^{14} + z^{13} - z^{12} - z^{11} - z^{10} - z^9 - z^8 + z^7 + z^6 - z^5 + z^4 - z^3 + z^2 + z - 1 = (z^{18} + z^{16} + 2z^{15} + 2z^{13} + z^{12} + 2z^{11} + z^{10} + 3z^8 + z^6 + 2z^5 + 2z^3 + 1)(z^2 + 1)(z - 1)(z^3 + z^2 - 1)^2$
* $z^{27} + z^{26} + z^{25} - z^{24} - z^{23} - z^{22} + z^{21} + z^{20} + z^{19} + z^{18} - z^{17} + z^{16} - z^{15} - z^{14} - z^{13} + z^{12} - z^{11} - z^{10} - z^9 - z^8 + z^7 + z^6 - z^5 + z^4 - z^3 + z^2 + z - 1 = (z^{18} + z^{16} + 2z^{13} - z^{12} + 2z^{11} + z^{10} + 3z^8 + z^6 + 2z^5 + 2z^3 + 1)(z^2 + 1)(z - 1)(z^3 + z^2 - 1)^2$


all have repeated factor $z^3 + z^2 - 1$ with root $z \approx 0.75488$.


