Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)

*To*: mathgroup at smc.vnet.net*Subject*: [mg57310] Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)*From*: Daniel Reeves <dreeves at umich.edu>*Date*: Tue, 24 May 2005 05:12:48 -0400 (EDT)*References*: <200505230620.CAA04045@smc.vnet.net> <EEA0FBBD-9C31-419A-8D0B-7C73AE4DA32E@akikoz.net>*Sender*: owner-wri-mathgroup at wolfram.com

Ooh ooh! I just got FindInstance to prove it for any particular n. (Tried it for all n up to 75 and it takes about 20 seconds each on my machine and doesn't slow down as n increases, though every once in a while there's an n that takes 2 minutes). n = 7; FindInstance[FullSimplify[D[f[x,n],x]] == 0 && 0<x<(n-1)/n, {x}, Reals] --> returns {} and so QED. Prize for the general case bigger than ever... > Unfortunately I can only manage the most trivial case, n=2, which I > do not think deserves any prize but it seems to show that if anyone > does get the prize it will be well deserved. > > We evaluate your definitions and then set > > n = 2; > > Even in this simplest of cases trying to use Solve directly > > > Solve[D[f[x,2],x]==0,x] > > $Aborted > > takes for ever, so we resort to Groebner basis (actually what I just > called "a hack" in another discussion ;-)) > > We first evaluate > > z = FullSimplify[D[f[x, 2], x] /. x^2 + 81 -> u^2, u > 0]; > > And then use GroebnerBasis: > > > GroebnerBasis[{x^2 + 81 - u^2, z}, {x}, {u}] > > {4*x^2 + 16*x - 9} > > This we can actually solve (even without Mathematica): > > > r=Solve[4*x^2 + 16*x - 9 == 0, x] > > {{x -> -(9/2)}, {x -> 1/2}} > > we can check that the derivative of f[x,2] does indeed vanish at > these points: > > > D[f[x,2],x]/.r//FullSimplify > > {0,0} > > Assuming that Groebner basis is correctly implemented this also shows > that f'[x,2] has no roots in the interval (0,1/2) so we are done. > > > Unfortunately the same method with n=3 already seems to take for > ever. For small n>2 it may be possible perhaps to combine in some way > numerical methods with Groebner basis to show similarly that D[f > [x,n],x] has no roots in the interval (0,(n-1)/n) but obviously this > won't work for a general n. At the moment the general problem looks > pretty intractable to me, so much that I am almost willing to offer > an additional prize to whoever manages to solve it (but I won't make > any offers before seeing other people's attempts ;-)) > > Andrzej Kozlowski -- http://ai.eecs.umich.edu/people/dreeves - - google://"Daniel Reeves" "The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell." -- St. Augustine

**References**:**Solve or Reduce on a monstrosity of an expresssion (and a prize!)***From:*Daniel Reeves <dreeves@umich.edu>