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

*To*: mathgroup at smc.vnet.net*Subject*: [mg57313] Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)*From*: Andrzej Kozlowski <andrzej at akikoz.net>*Date*: Tue, 24 May 2005 05:12:52 -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>*Sender*: owner-wri-mathgroup at wolfram.com

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 > >

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