Reduce[y <= x^(1/4) && y >= x^2, {x, y}, Reals] outputs (x == 0 && y == 0) || (0 < x < 1 && x^2 <= y <= Root[-x + #1^4 &, 2]) || (x == 1 && y == 1)
When I try to rewrite the root, I want the answer seems to be incorrect:
Reduce[y <= x^(1/4) && y >= x^2, {x, y}, Reals] // ToRadicals outputs (x == 0 && y == 0) || (0 < x < 1 && x^2 <= y <= -x^(1/4)) || (x == 1 && y == 1), but the correct answer is x^2 <= y <= x^(1/4).
As I explained in the comment - with ToRadicals "The result may not be equal to the Root object for some values of the parameter".
I do not know why but if you omit the variables in Reduce then the output is different (without the cumbersome Root). Or the same if you reverse the order of variables from {x, y} to {y, x}.
Reduce[y <= x^(1/4) && y >= x^2, {x, y}, Reals]
Reduce[y <= x^(1/4) && y >= x^2, {y, x}, Reals]
Reduce[y <= x^(1/4) && y >= x^2, Reals]
(x == 0 && y == 0) || (0 < x < 1 &&
x^2 <= y <= Root[-x + #1^4 &, 2]) || (x == 1 && y == 1)
(y == 0 && x == 0) || (0 < y < 1 && y^4 <= x <= Sqrt[y]) || (y == 1 &&
x == 1)
(y == 0 && x == 0) || (0 < y < 1 && y^4 <= x <= Sqrt[y]) || (y == 1 &&
x == 1)
Reduce, the order of variables matter: Reduce[expr, {x1, x2, ...}] effectively writes expr as a combination of conditions on x1, x2, … where each condition involves only the earlier xi.
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Reduce produces unnecessary complicated expression y <= Root[-x + #1^4 &, 2] instead of straightforward expression y <= x^(1/4) while for another order of variables this does not happen. It could produce x <= Root[-y + #^2 &, 1] instead of x <= Sqrt[y] similarly like with original ordering of variables.
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Commented
2 hours ago
Root[-x + #1^4 &, 2] // ToRadicals gives -x^(1/4) and Root[-x + #1^4 &, 4] // ToRadicals gives x^(1/4). Possibly the problem is that Reduce gives an incorrect root.
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Reduce gives the correct root, which can be verified by plotting. It is ToRadicals that select a different root but it cannot be taken as a bug because it is mentioned in the documentation of possible issue of ToRadicals.
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Commented
1 hour ago
ToRadicals. If the root contains parameter "The result may not be equal to the Root object for some values of the parameter". But what is strange is whyReduceproduced cumbersome resulty <= Root[-x + #1^4 &, 2]instead ofy <= x^(1/4). $\endgroup$