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In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals.

Here "nested radicals" means expression such as: $$ \sqrt[n]{a+b\sqrt[n/p]{a+b\sqrt[n/p]{a+b \cdots}}} $$ i.e. with infinite radicals nested each other.

This means that every algebraic number can be expressed by a sort of ''generalized radicals'' and, since every such number can also be expressed by a series, I've searched if there is some way to transform infinitely nested radicals into series. Searching on the web I've find nothing interesting, so my question is:

there is some canonical way to transform an infinitely nested radical in a series?

Das
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Emilio Novati
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  • A one minute glance at the paper didn't find any claim to solve "any polynomial equation". Where did you see that? – Bill Dubuque Mar 05 '15 at 20:44
  • Abstract anf final statement pag 7 – Emilio Novati Mar 05 '15 at 20:49
  • But also I'm re-reading the paper to see if the proof is correct. Anway the question has some sense – Emilio Novati Mar 05 '15 at 20:53
  • @EmilioNovati: The paper is interesting, but needs some refining. The general 5th deg is known to be solved in infinitely nested radicals. Bagis's similar result for the general 6th deg apparently is new. However, his results for 7th deg and higher seem to be hand-waving. His eqs 22-30 are problematic and he even mentions in a note about "conjectures". Some claims need to be clarified. – Tito Piezas III Mar 06 '15 at 03:29
  • @Tito: I see. My word ''proved'' in OP may be too strong, so I've edited. But the reference was only a context for my question. Anyway i'm interesting about the possibility to transform an infinitely nested radicals in some kind of series. – Emilio Novati Mar 06 '15 at 10:52

1 Answers1

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...canonical way to transform an infinitely nested radical in a series?

There is no such way in general. In fact, general non-periodic nested radicals allow no alternative ways to express them (except for trivial identity transformations, which still lead to a nested radical expression).

If we are talking about periodic nested radicals, connected to polynomial equations, such as in the OP, it might indeed be possible to express their limit as a series (if they converge). But as far as I know, there is no better way to do that, than to consider the original algebraic equation.

The theory of nested radicals still has a long way to grow. There is not much known about them, except for the convergence theorem. I suggest reading this article which gives a very good review on the topic of nested radicals and other such expressions.

Also, there are two papers by Dixon Jones, investigating a more general case of continued powers (which also incorporates infinite series and products, continued fractions, etc):

His papers can be found here: first and second.

Mathworld entry here provides other references and examples.

Yuriy S
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