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Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by $$ f(x)=x^q-kx^{q-p}-\ell. $$ This polynomial is related to a family of two-parameters binary sequences.

Now, I would like to prove that $f$ has a dominant root and that all the roots are simples. I was able to solve for some particular cases (e.g., $p=2$ and $k=q=1$).

Someone has some suggestion for proving this, in the general case? Thanks in advance.

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  • $\begingroup$ if $q$ and $p$ are both divisible by 3, then with each root $\theta$ there are roots $e^{2\pi i/3}\theta$ and $e^{-2\pi i/3}\theta$, which of them is "dominant"? $\endgroup$
    – Fedor Petrov
    Commented Jan 24, 2020 at 20:24
  • $\begingroup$ Thanks for your comment. In fact, I forgot to say that, clearly, $p$ and $q$ are coprime. $\endgroup$
    – Arthut
    Commented Jan 24, 2020 at 20:40
  • $\begingroup$ That assumption seems to render "$f$ is not an even function" redundant, no? $\endgroup$
    – user44191
    Commented Jan 24, 2020 at 20:46
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    $\begingroup$ Sorry I keep adding and deleting an answer, but I keep noticing small different criteria that I think I missed. I do think that your claim of simple roots is incorrect; integer solutions to $\ell^p = \frac{(q - p)^{q - p} k^q p^p}{q^q}$ for a given even $p$, odd $q$ will imply a double root. $\endgroup$
    – user44191
    Commented Jan 25, 2020 at 2:53
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    $\begingroup$ For an example: $x^5 - 15x^3 - 162$ has a double root at $-3$. $\endgroup$
    – user44191
    Commented Jan 25, 2020 at 5:48

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