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."

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

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

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