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

*To*: mathgroup at smc.vnet.net*Subject*: [mg57420] Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)*From*: Daniel Reeves <dreeves at umich.edu>*Date*: Fri, 27 May 2005 04:57:08 -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> <Pine.LNX.4.58.0505241736010.24188@boston.eecs.umich.edu> <1D91709F-1E4D-4B98-95F5-695F7BD65577@akikoz.net> <Pine.LNX.4.58.0505242049410.3527@boston.eecs.umich.edu> <F016112F-BF6B-4D4E-8A3C-1F38AB3DDC5F@akikoz.net> <Pine.LNX.4.58.0505251303210.3527@boston.eecs.umich.edu> <4294C165.8040406@wolfram.com>*Sender*: owner-wri-mathgroup at wolfram.com

I'm taking a few parts on faith (I guess technically we all are given the computer-assisted nature of this proof!) but it sounds like you and Andrzej have similar ideas about establishing at most 2 roots. I don't know what you have in mind though for how to establish "sufficiently many" values of n. n=2,3,4,5,6, and 7 all work but for n=8, GroebnerBasis does not include a degree 2 polynomial. (Still, FindInstance confirms no roots in (0,(n-1)/n) for n=8, and in fact does so all the way up to 684.) --- \/ FROM Daniel Lichtblau AT 05.05.25 13:18 (Today) \/ --- > Daniel Reeves wrote: > > yikes. ok, let's assume that all works. So it remains to show that both > > of the following expressions of n are zero for all n >= 2. Strangely, > > FullSimplify can do this for all specific n up to 113 but 114 seems to > > stump it. > > > > here are the expressions: > > [...] > > > I can point out some aspects to a probabilistic argument. Making it into > a full fledged proof would require figuring out some bounds that offhand > I don't know how to do, then verifying for appropriately many cases. > > First you need to show that there are 2 roots for each x, at least when > n>=2. For this it suffices to show it holds for generic n. Since we > cannot seem to do that computationally, instead do it for "random" n. > Calling your expression fp, do, say, > > fp7 = Simplify[fp /. n->7] > gb7 = GroebnerBasis[fp7,x] > > Now check for a quadratic polynomial (in x) in gb7. If you find one, > that means there are at most 2 solutions when n=7. This can of course be > repeated for other values of n. The point is that, for integers larger > than 2, this computation appears to be feasible. Offhand I think this > idea (of verifying at sufficiently many values of n) suffices, though > there may be a lurking branch cut issue. Note that showing there are two > solutions for n=7 is not quite sufficient in and of itself because 7 > might not be generic. In this setting, nongeneric points would lie on a > variety (roots of a polynomial), hence be a set of measure zero. > > Verification that x->(n-1)/n and x->-(n+1)^2/(n*(n-1)) are the two > solutions can be done by making the substitutions into fp and verifying, > for sufficiently many n>=2, that these vanish. The reason is that > univariate algebraic functions vanishing at sufficiently many integer > points vanish everywhere that branches don't change (e.g. x-Sqrt[x^2] > vanishes on positive integers and indeed on all positive reals). I think > the branch cut issue goes away when you have the proviso that n>=2, > because for the substitutions above all radicands are positive in that > region. But maybe I'm missing something and there is more to it than this. > > I should mention that if you did not know those were the appropriate > substitutions, they might be deduced by finding particular {x,n} pairs > and getting more values via homotopy continuation (basically, setting up > and solving an ODE via NDSolve). Values from this could then be used to > interpolate an implicit relation. This was the gist of some work I > described in a note last week: > > http://forums.wolfram.com/mathgroup/archive/2005/May/msg00627.html > > Offhand I do not know how well it would work for your particular > algebraic expression, but it strikes me as a plausible approach. (A > better one for this case might be to take an implicit plot of the zero > set of fp and guess the relation). > > > Daniel Lichtblau > Wolfram Research > -- http://ai.eecs.umich.edu/people/dreeves - - google://"Daniel Reeves" "To me, boxing is like a ballet, except there's no music, no choreography, and the dancers hit each other." -- Jack Handy, Deep Thoughts

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

**Re: plot hyperbola (OT)**

**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!)**

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