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Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5?

More specifically: does there exist any ''reasonable'' sequence of ''reasonable'' finite sets of ''reasonable'' special functions such that for arbitrary polynomial of degree d its roots are expressible by a unique term made out of coefficients of polynomial and functions from the d-th set?

By ''reasonable'' special functions I mean ones that has been investigated independently or have been already coined and are decently understood or are solutions to reasonable ODEs.

Are there some general results on restrictions to such sequence of special functions? Existence of integer solutions to polynomial equations is an undecidable problem. So if we replace ''reasonable'' by ''recursive'' and define ''reasonable special function'' as one defined by a power series with rational coefficients $(a_n)$ computably depending on $n$, then I imagine the answer might be 'no'. But no obvious proof comes to my mind that such a family may not exist.

user57888
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    It would be nice if the closers gave some more specific reasons why this was closed. The question as expressed in the first paragraph seems quite reasonable to me, and it would be good to get an expert response. In the meantime, perhaps the text Beyond the Quartic Equation by R. Bruce King has some useful information. – Todd Trimble Sep 05 '14 at 19:14
  • I voted to close this question as "unclear" several versions ago. I have voted to reopen since the question now states clearly which method or result is to be generalized. – Ricardo Andrade Sep 05 '14 at 23:45
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    It seems like the topic of this question is amply covered by http://math.stackexchange.com/questions/291909 and http://mathoverflow.net/questions/23094 – j.c. Sep 06 '14 at 10:35
  • Great! Thank you for the book reference and link. I only browsed through MO and tried to google it. As a response to the same question posted in MS one of the comments to the following question has been linked: http://scicomp.stackexchange.com/questions/1864/why-cant-householder-reflections-diagonalize-a-matrix/1885#1885 – user57888 Sep 06 '14 at 10:50
  • Other previous questions http://mathoverflow.net/questions/89144/can-roots-of-any-polynomial-be-expressed-using-eulerian-function http://mathoverflow.net/questions/61409/using-higher-order-bring-radicals-to-solve-arbitrary-polynomials . – David E Speyer Sep 08 '14 at 14:05
  • Thank you. I haven't originally found the first question and the second seemed to specialized for me. In particular I wasn't able to tell, whether the question intended was: ''We know that roots of arbitrary polynomials are expressible in terms of some class of special functions. Can it be also done with Bring radicals?'' or rather ''We do not know much about finding roots of arbitrary polynomials. Are Bring radicals a solution to this problem?'' – user57888 Sep 09 '14 at 20:41

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