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Unexpected failures of FullSimplify on some Root elements


For some reason FullSimplify does not resolve some of the Root elements
while it does others.
In this example Root[2 - 4 #1^2 + #1^4 &, 2] and Root[6 - 6 #1^2 + #1^4 &,
2] go unresolved when you evaluate:

Function[k, FullSimplify[Root[Function[x, k - (k - x^2)^2], #] & /@
Range[4]]] /@ Prime[Range[11]]

This is despite the fact that Solve[k - (k - x^2)^2 == 0, x] can do so in
general.

It seems that the second root is most prone not the be resolved.
Here Root[-2 - 12 #1^2 + 9 #1^4 &, 3] also appears when you evaluate:

Function[k, FullSimplify[Root[Function[x, k - (k - x^2)^2], #] & /@
Range[4]]] /@ (Prime[Range[11]]/3)

In this more involved example still only Root[2 - 4 #1^2 + #1^4 &, 2] goes
unresolved, try evaluating:

FullSimplify[Root[Function[x, 2 - (2 - (2 - (2 - x^2)^2)^2)^2], #] & /@
Range[16]]

Strangley enough, Mathematica appears to know the resolution Root[2 - 4 #1^2
+ #1^4 &, 2] is -Sqrt[2 - Sqrt[2]] even though this result is not returned,
try evaluating:

Trace[FullSimplify[Root[2 - 4 #1^2 + #1^4 &, 2]], TraceInternal->True,
TraceOriginal->True, TraceDepth->2]

Note that the result is cached, so you will see a great deal more Trace
output if you run this command as the first thing after restarting
Mathematica.

By the way, is this fixed in version 6.0.2?

Q.E.D.





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