Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at least from what I skimmed over.
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2 Answers
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Do you ask about solution to high degree polynomials? The following are some reference:
Umemura H. (2007) Resolution of algebraic equations by theta constants. In: Tata Lectures on Theta II. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. doi:10.1007/978-0-8176-4578-6_18
L. Bers, Uniformization, Moduli, and Kleinian Groups, Bulletin of London Math. Soc. 4 (1972) 257-300, doi:10.1112/blms/4.3.257
F. Klein, Lectures on the icosahedron, 2nd edition, Dover, 1956; Internet Archive
Prasolov and Solovyev, Elliptic functions and elliptic integrals, Translations of Mathematical Monographs 1997. AMS bookstore
Richard J. Drociuk, On the Complete Solution to the Most General Fifth Degree Polynomial https://arxiv.org/abs/math/0005026
answered Jun 16, 2013 at 11:29
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$\begingroup$ Let me just comment that the Drociuk paper writes the solution in terms of a special value of a hypergeometric function, which is essentially a Bring radical. It doesn't claim to break the Abel–Ruffini theorem, despite the alarm bells the appearance (and some of the prose) of the preprint might raise. $\endgroup$– David Roberts ♦Commented Jan 4, 2022 at 6:12
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There's also a very nice, modern exposition by Mark L. Green: On the analytic solution of the equation of fifth degree, Compositio Mathematica 37 (1978), 233–241. A pdf of the paper is available online: http://www.numdam.org/item?id=CM_1978__37_3_233_0
answered Jun 16, 2013 at 12:57