That big sextic is "the" resolvent of the septic 8903 + 47647 v + 39672 v^2 + 7192 v^3 - 522 v^4 - 174 v^5 + v^7 that Tito used to solve x^8-x^7+29*x^2+29, which at least "explains" the 29s. --rwg On Tue, Sep 20, 2011 at 3:06 PM, <jdb@math.arizona.edu> wrote:
Bill Gosper <billgosper@gmail.com> wrote:
On Tue, Sep 20, 2011 at 3:34 AM, Bill Gosper <billgosper@gmail.com> wrote:
RootAp1[z_, n_] := Block[{r = RootApproximant[z, n]}, If[Log[Abs[(MinimalPolynomial[r] /. Plus -> Times)@1]] > 9*Precision[z], 1, r]]
[This replaces previous version, which had one of those delicious bugs that disappears when you insert Print statements. Thank you for not playing with it and sending me hate mail.-]
Ssexy[q_]:=Catch[Block[{rts,foos=1,v=Variables[q][[1]],Q},rts=#[[1,2]]&/@NSolve[q==0,WorkingPrecision->Ceiling[Log[Abs[Discriminant[q,v]]]]];If[FreeQ[Q=Solve[0==q],Root],Q,
Do[foos*=Select[v*Factor[q,Extension->{RootAp1[rts[[j]]+rts[[k]],3],RootAp1[rts[[j]]*rts[[k]],3]}],Exponent[#,v]==2&];If[Exponent[foos,v]==6,Throw[ToRadicals[Solve[0==foos]]]],{j,5},{k,j+1,6}];foos=Factor[q];
Do[If[foos=!=(foos=Factor[q,Extension->(RootAp1[#,2]&/@CoefficientList[(#-rts[[i]])*(#-rts[[j]])*(#-rts[[k]]),#])]),Throw[Solve[foos==0]]],{i,4},{j,i+1,5},{k,j+1,6}]];Q]]
This is seriously untested, longer, faster on most cases, but several minutes on Ssexy[2925951033851274156588135512485165232256823853056 - 2697817290737324449800236848640264992467435520 #1 + 9932351343021963689693473396732415411860992 #1^2 - 1881654619801628210127689611068977299937 #1^3 + 11450425009897563891465337536606118710 #1^4 + 3847649781964086608961673413540069 #1^5 + 378818692265664781682717625943 #1^6]
JDB>
Bill, I noticed the constant term beginning 29259... is actually equal to 2^21*29^21*43^7 and wondered why these three primes play such a role in all these coefficients.