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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
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