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Radical Denest: an ancient difficult task in symbolic computation

Posted 1 year ago
3 Replies
12 Total Likes

3 Replies
Posted 1 year ago

Very good! Is there another similar function that does similar things when variables (rather than just numbers) are involved?

For example, there is the function

denestSqrt[e_, domain_, x_] := Replace[y /. Solve[Simplify[Reduce[Reduce[y == e && domain, x], y,
   Reals],  domain], y], {{r_} :> r, _ -> e}]

written by @CarlWoll (denestSqrt).

Here is an example:

denestSqrt[Sqrt[(1 - 2 x) (1 - x - 2 x^2) (2 - x + 2 Sqrt[1 - x - 2 x^2])], 1/3 < x <= 1/8 (2 + Sqrt[2]), x]
(* 1 - x - 2 x^2 + Sqrt[-(1 + x) (-1 + 2 x)^3] *)

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Posted 6 days ago

Looks fantastic, this is just what I need for recent work. However, the current algorithm is unable to find some denesting result even when one truly exists, e.g. ones related to non-algebraic numbers and transcendental functions, like this below:

$$ \frac{1}{2} \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}} $$

From my test today, the function posted is unable to find the denesting result of the compound radicals above, but there truly exists one:

$$\frac{1}{2} \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}}=\sin \left(\frac{\pi }{96}\right)$$

So I hope you can improve your function, making it able to solve compound radicals related to non-algebraic numbers and transcendental functions.

Despite its imperfection, I am still very impressive about Corey Ziegler and Bill Gosper, Daniel Lichtblau, Swastik Banerjee and their team's excellent work. This is no doubt a good start. Please keep going, making this function more and more powerful!

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