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Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)
*To*: mathgroup at smc.vnet.net
*Subject*: [mg57316] 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 05:12:54 -0400 (EDT)
*References*: <200505230620.CAA04045@smc.vnet.net> <EEA0FBBD-9C31-419A-8D0B-7C73AE4DA32E@akikoz.net> <Pine.LNX.4.58.0505231817420.9452@boston.eecs.umich.edu> <458D701E-37FA-425F-89C4-52A5628E22CF@akikoz.net>
*Sender*: owner-wri-mathgroup at wolfram.com
I believe your conjecture with all my heart (in case you missed my other
message, I inferred the 2 roots for general n based on empirical evidence
and they in fact match the roots of your quadratic below).
But I'm at a total loss for how to prove it!
--- \/ FROM Andrzej Kozlowski AT 05.05.24 10:15 (Tomorrow) \/ ---
> Actually I have now found that
>
>
> GroebnerBasis[FullSimplify[D[f[x, n], x]], x]
> also works for n other than 2 and confirms my conjecture. For
> example;
>
>
> GroebnerBasis[FullSimplify[D[f[x, 3], x]], x]
>
>
> {-9*x^2 - 18*x + 16, 9*x + 18 - 16/x,
> -18*x - 9*(x^2 + 54) + 502,
> 27*6^(1/3)*Sqrt[x^2 + 54]*x + 35*6^(5/6)*x -
> 68*6^(1/3)*Sqrt[x^2 + 54] + 350*6^(1/6)*
> (x^2*(Sqrt[6]*Sqrt[x^2 + 54] + 18))^(1/3) -
> 140*6^(5/6)}
>
>
> Solve[-9*x^2 - 18*x + 16 == 0, x]
>
>
> {{x -> -(8/3)}, {x -> 2/3}}
>
> I suspect that this is actually what FindInstance uses (?)
>
>
> So here is a new formulation of the conjecture: the roots of the
> equation D[f[x,n],x]==0 are precisely the roots of the quadratic
> equation
>
>
> x^2 + (n*x)/(n - 1) + (2*x)/(n - 1) + x/((n - 1)*n) +
> x/n - x - n/(n - 1) - 1/(n - 1) + 1/((n - 1)*n) +
> 1/((n - 1)*n^2) ==0
>
>
> Andrzej Kozlowski
>
>
>
> On 24 May 2005, at 08:11, Daniel Reeves wrote:
>
> > 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
> >
> >
>
--
http://ai.eecs.umich.edu/people/dreeves - - google://"Daniel Reeves"
"To err is human. To moo bovine."
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