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I can't seem to find a good resource on Vieta's root jumping, which would explain various scenarios that suggest using the technique.

Does anyone have a suggestion for a reference?

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  • $\begingroup$ These are special cases of results on Pell equations, or equivalent results in quadratic fields (e.g. Richaud-Degert quadratics which have short continued fractions so small fundamental units). See Bill Dubuque's remarks in this question. $\endgroup$ – Math Gems Jan 28 '13 at 20:35
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EDIT: start with THIS

The nontrivial use of this is in the Markov spectrum. See CUSICK and FLAHIVE. Kap and I found a number of other trees and related problems, see KAP PDF . Hurwitz expanded Markov's original tree in three variables to $n$ variables about 1907, see info in https://mathoverflow.net/questions/84927/conjecture-on-markov-hurwitz-diophantine-equation . The result becomes a forest rather than a single tree for some $n \geq 14.$ Many related problems are possible, as the term $a x_1 x_2 \ldots x_n$ can be replaced by any homogeneous symmetric polynomial as long as the exponent on each $x_i$ is 1 and the total degree of each term is at least 3.

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  • $\begingroup$ Thanks, this looks interesting! $\endgroup$ – Calvin Lin Jan 29 '13 at 8:44
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Yimin Ge has written a short note, http://www.yimin-ge.com/doc/VietaJumping.pdf

Project Pen also had a short piece on IMO 1988 number 6, http://projectpen.files.wordpress.com/2008/10/pen-vol-i-no-1.pdf

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    $\begingroup$ I've seen both of those. I'm looking for reasons to motivate using of Vieta jumping. For example, while the technique not limited to quadratic equations, I have yet to see one which involves a cubic. Is there any reason why VJ isn't applied to higher degree polynomials (apart from having multiple roots)? $\endgroup$ – Calvin Lin Jan 28 '13 at 6:56
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    $\begingroup$ @CalvinLin, it seems hard to motivate an Olympiad trick, which doesn't even show up that much. (Appearance in IMO 2007 already attracted criticism) Not sure if higher degree polynomial can enter the picture. $\endgroup$ – user27126 Jan 28 '13 at 7:00
  • $\begingroup$ @CalvinLin: I have been experimenting with cubic and quartic Vieta jumping. I would be happy to discuss my results with you off-line, and then bring any meaningful results back onto MSE. $\endgroup$ – Kieren MacMillan Jul 5 '14 at 15:18
  • $\begingroup$ @KierenMacMillan That's great. Can you send me an email at calvin@brilliant.org? I would be interested in looking through it. $\endgroup$ – Calvin Lin Jul 5 '14 at 15:28

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