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It is proved here that the equation $1+z+z^2=z^n$ have no multiple complex roots.

Q. Let us consider the equation $1+z+z^q=z^n$ where $q$ and $n$ are natural numbers with $1<q<n$. Any characterization for the pair $(q,n)$ such that the equation $1+z+z^q=z^n$ have no multiple complex roots?

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1 Answer 1

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The roots are always simple, this follows without further reasoning from the old paper On the irreducibility of certain trinomials and quadrinomials by Ljunggren. Set $f(z)=z^n-1-z^p-z^q$. Then Ljunggren proves (Theorem 1) that $f(z)=Q(z)g(z)$, where the roots of $Q$ are roots of unity (so $Q=1$ if there are no such roots), and where $g(z)\in\mathbb Q[z]$ is irreducible none of its roots is a root of unity. As an irreducible polynomial, $g$ has no multiple roots. Furthermore (Theorem 2), Ljunggren shows that $Q$ has simple roots either, which settles the question.

Remark: In case Ljunggren's paper is not accessible, this review at zbMATH contains the full statement of Ljunggren's results needed here.

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  • $\begingroup$ Great! It is so helpful. $\endgroup$
    – ABB
    May 20, 2023 at 11:10
  • $\begingroup$ I tried to examine the problem for the equations $1+z^p+z^q+z^r=z^n$. But seems one can not follow that by Ljunggren's paper! Since I could not find any other idea to figure it out, it is asked in a new problem. $\endgroup$
    – ABB
    May 25, 2023 at 18:33
  • $\begingroup$ Could we consider the following as an alternative answer? If there is a multiple root, it should be satisfied $nz^{n-1}-pz^{p-1}-qz^{q-1}=0$. By a combination with the original equation, ($z^n-1-z^p-z^q=0$) we may conclude that $(n-p)z^p+(n-q)z^q-n=0$. But this latter (of degree $p$) divides no polynomial form $\mathbb{Z}[x]$ meaning that the original polynomial $z^n-1-z^p-z^q=0$ can not have multiple roots! (We are using Gauss lemma) $\endgroup$
    – ABB
    Nov 8, 2023 at 17:05
  • $\begingroup$ @ABB For this to work you would need to know that your degree $p$ polynomial does not have a monic divisor over $\mathbb Z$. $\endgroup$ Nov 8, 2023 at 17:47
  • $\begingroup$ @ Peter Meuller, if $z^p+\frac{n-q}{n-p}z^q-\frac{n}{n-p}$ divides a monic polynomial from $\mathbb{Z}[x]$ then the coefficients are all integer which is impossible. $\endgroup$
    – ABB
    Nov 8, 2023 at 18:05

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