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Can a polynomial of the form $\sum\limits_{k=0}^n a_k x^k$, where $a_k=\pm1$, have arbitrarily many distinct real roots?

I suspect the answer is yes. So far, the most roots of such a polynomial that I have found, is five: $x^{13}+x^{12}-x^{11}-x^{10}-x^9-x^8-x^7+x^6+x^5-x^4+x^3-x^2+x+1$.

(This question was inspired by a question, "What is the maximum number of distinct real roots of a polynomial whose coefficients are all $1$, $-1$ or $0$?")

Dan
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    Doesn't Sil's construction of $f(x) = x^2 - x - 1$ in the linked problem work directly? You can check that $f(x), f(x) f(x^3) = g(x), g(x)g(x^9)=h(x), \ldots$ works. Alternatively, take $ f(x), f(x) f(x^3), f(x)f(x^3)f(x^9), \ldots$, where the coefficient could be figured out by looking at the ternary expansion. – Calvin Lin Apr 11 '23 at 05:56
  • @CalvinLin Ah, yes! I didn't realize that all the coefficients in Sil's construction starting with $f(x) = x^2 - x - 1$ are all $\pm1$ (no $0$ coefficients). That answers my question. – Dan Apr 11 '23 at 06:15
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    @Gonçalo What I mean is, for any $m\in\mathbb{N}$, there exists a polynomial of the form $\sum_{k=0}^n a_k x^k$ where $a_k=\pm 1$, such that the polynomial has at least $m$ distinct real roots. – Dan Apr 11 '23 at 06:17

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From Sil's answer to another question, and Calvin Lin's comment:

Let $f(x)=x^2-x-1$.

The polynomial $g(x)=f(x)f(x^3)f(x^9)...f(x^{(3^m)})$ is of the form $\sum\limits_{k=0}^n a_k x^k$ where $a_k=\pm1$.

By increasing $m$, $g(x)$ can have arbitrarily many distinct real roots (as proved by Sil's answer).

Dan
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  • Why does Sil's answer say -1, 0, 1 when only coefficients $\pm 1$ are used? – Snared Apr 11 '23 at 07:15
  • My construction was generic for $-1,0,1$ polynomials, we could have started with $x^3-x-1$, I haven't noticed that the example actually only requires $\pm 1$, that is nice catch. Also in this example it's easy to actually count and list all the roots, out of which there are precisely $2m+2$ distinct real roots given by $\sqrt[3^n]{\frac{1\pm \sqrt{5}}{2}}$ for $n=0,1,\dots,m$ (roots taken in real numbers). – Sil Apr 12 '23 at 14:34