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I know about Abel–Ruffini theorem, but I have a polynomial of special form. From "Beyond the Quartic Equation" by R.B. King (a very interesting book, btw) I've learned about Tschirnhaus transformations which I try to use, to convert my polynomial $$ x^9 + ax^6 + bx^5 + cx^3 + d = 0$$ to the form $$ x^9 + ax^6 + bx^3 + c = 0, $$ so I could do substitution $t = x^3$ and use Cardano's formulas. What other things I can try?

I would prefer solution in radicals, but closed form solution with elliptic functions will also be satisfying.

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    $\begingroup$ I would say it is not possible with Tschirnhaus transformations, because to get rid of all powers except multiples of 3, you need many parameters, so degree of polynomial involved in transformation will be greater than 5, so unsolvable. That doesn't say anything about specific cases though. $\endgroup$
    – Somnium
    Commented Aug 5, 2022 at 15:37

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There is no way to transform your first polynomial to the special shape of the second polynomial while preserving its Galois group. The Galois group of the second polynomial is solvable, but for instance the Galois group of $f=X^9+X^5+1$ is the full symmetric group $S_9$ (as can be seen by the factorizations modulo $2$, $3$ and $5$, respectively).

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  • $\begingroup$ Do I have some options of approximating original polynomial if I allow roots of new approximate polynomial to be in epsilon neighborhood of the original one? To be more specific I care only about one specific root. $\endgroup$
    – Moonwalker
    Commented Aug 8, 2022 at 12:51

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