Unexpected failures of FullSimplify on some Root elements

*To*: mathgroup at smc.vnet.net*Subject*: [mg86538] Unexpected failures of FullSimplify on some Root elements*From*: "Q.E.D." <aoe at netzero.net>*Date*: Thu, 13 Mar 2008 04:33:27 -0500 (EST)*Organization*: Adtech Computers, Inc.

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.

**Follow-Ups**:**Re: Unexpected failures of FullSimplify on some Root elements***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>