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There might not be a known generalized inverse for all polynomials, but it seems like there should be some kind of generalization for certain kinds of simple polynomials, like all polynomials in the form of f(x)=(x+n)^x or f(x)=x^a+x^b or polynomials of a similar simplicity. I don't necessarily know what they are, but the cubic equation for instance was discovered in near the 17th century and I figured there must be some kind of progress on this front after this many decades.

I don't mean a generalized solution for just any one specific degree of polynomial at a time. I'm not looking for a quintic formula and then a hextic formula and then a septic formula and etc, I'm looking for one formula for all degrees of polynomials that have a specific but simpler structure.

And even though the inspiration stems from simple algebra, I am still completely open to techniques in complex analysis or more advanced algebra and functional analysis if there are any.

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  • $\begingroup$ Well it is useful for example here: en.wikipedia.org/wiki/…, where a strict characterization of which quintics are algebraically solvable and which aren't. This is uses a lot of abstract algebra. if this is irrelevant, let me know, i may have misunderstood your intent $\endgroup$ Commented Apr 22, 2017 at 1:56
  • $\begingroup$ Yes if I was asking for some kind of proof of the Albel-Ruffini theorem that would be useful, but ideally the solution would involve algebra in nature, we're talking about a generalization of all polynomials of any chosen degree but that are limited to a very specific form, so knowing that a generalized solution containing only radicals and elementary operations isn't possible for degrees above 4 wouldn't be useful anyway. I'm not really looking for a proof, so if a solution for a certain kind of polynomial does contain complex numbers, I wouldn't be immediately concerned with how it works. $\endgroup$
    – RayOfHope
    Commented Apr 22, 2017 at 1:58
  • $\begingroup$ That link basically says that there is a closed form for all polynomials of the form $x^5 + ax + b$ IF it meets the condition in the link. And perhaps in the papers they give the actual formula to solve it. Generally, looking at very large families of solvable polynomials, and creating a closed form solution for them will use abstract algebra heavily. But if you want to avoid its use thats also fine. $\endgroup$ Commented Apr 22, 2017 at 2:01
  • $\begingroup$ And it's great that it says that, but like, what about 6th degree polynomials and 7th degree polynomials and 8th degree polynomials? $\endgroup$
    – RayOfHope
    Commented Apr 22, 2017 at 2:02
  • $\begingroup$ Yea so that will be even more complicated machinery than the stuff in the link in the general case, possibly even still an open problem. Let me dig around and see if i can find something helpful to you $\endgroup$ Commented Apr 22, 2017 at 2:03

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This is a remarkably huge question depending on what you mean by "certain kinds of simple polynomials," but perhaps cyclotomic polynomials will satisfy you.

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  • $\begingroup$ That is more along the lines of what I'm looking for, though there's no reason to stop there and I don't see an immediately clear inverse, so thank you for that. It might seem like a huge question if you think about all polynomials, but if you stick to seeking out only limited but varying forms of polynomials like these cyclotomic polynomials, it's much less mind boggling. If 10 different people had 10 different small polynomial generalizations, that would suffice. $\endgroup$
    – RayOfHope
    Commented Apr 23, 2017 at 8:17

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