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From: F/32 Eurydice on 15 Jun 2010 19:24 What's the highest degree polynomial for which all roots can be written in terms of known functions? TIA.
From: Rick Decker on 15 Jun 2010 19:51 On 6/15/10 7:24 PM, F/32 Eurydice wrote: > > What's the highest degree polynomial for which all roots can be > written in terms of known functions? TIA. x^n, for n = 1, 2, ... ? Regards, Rick
From: Frederick Williams on 16 Jun 2010 12:20 F/32 Eurydice wrote: > > What's the highest degree polynomial for which all roots can be > written in terms of known functions? TIA. I think that roots can be written as functions of coefficients for all degrees: theta functions do for the lot. -- I can't go on, I'll go on.
From: F/32 Eurydice on 16 Jun 2010 13:15 On Jun 16, 12:20 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > F/32 Eurydice wrote: > > > What's the highest degree polynomial for which all roots can be > > written in terms of known functions? TIA. > > I think that roots can be written as functions of coefficients for all > degrees: theta functions do for the lot. > > -- > I can't go on, I'll go on. Can you give me a convenient reference for this?
From: achille on 16 Jun 2010 18:47
On Jun 17, 1:15 am, "F/32 Eurydice" <f32euryd...(a)sbcglobal.net> wrote: > On Jun 16, 12:20 pm, Frederick Williams > > <frederick.willia...(a)tesco.net> wrote: > > F/32 Eurydice wrote: > > > > What's the highest degree polynomial for which all roots can be > > > written in terms of known functions? TIA. > > > I think that roots can be written as functions of coefficients for all > > degrees: theta functions do for the lot. > > > -- > > I can't go on, I'll go on. > > Can you give me a convenient reference for this? In 1870, Jordan first showed any algebraic equation can be solved using modular functions[1]. In 1984, Umemura Hiroshi has written down an explicit basis for expressing the roots on any algebraic equation by higher genus theta functions[2]. [1] C.Jordan - Trait'e des Substitutions et des E'quations Alge'briques, Gautheirs-Villars, Paris 1870. [2] H.Umemura - Resolution of Algebraic equations by Theta Constants. appear as an appendix in the book "Tata Lectures on Theta II: Jacobian theta functions and differential equations", D. Mumford, ed. pp, 3.261-272, Birkhauser, Boston, 1984. BTW, I copied these reference from R. Bruce King's book "Beyond the Quartic equation (Modern Birkhauser Clasics)". It is a book for non-specialist which discuss how to solve quintic polynomials and beyond. |