On Sat, Aug 31, 2013 at 10:33 AM, Noam Elkies <elkies@math.harvard.edu>wrote:
Bill Gosper <billgosper@gmail.com> writes:
Wow, 78 roots, two real. Here's one of those: [...]
A trinomial can never have more than two real roots (Descartes' rule).
With what disclaimer? 4 x - 5 x^3 + x^5 has five. rwg>In[104]:= N[%%%^6 + 49*%%%, 49] Out[104]= 49.00000000000000000000000000000000000000000000000 (RootReduce wimped out.) Likewise MinimalPolynomial. But at least In[161]:= Timing[ MinimalPolynomial[ 67/246 Sqrt[7] Sin[1/3 ArcCsc[2 Sqrt[7]]] + 245/738 Sin[1/3 ArcSin[71/98]] + 469/738 Sin[1/3 (4 \[Pi] - ArcSin[13/14])] + 35/246 Sin[1/3 (4 \[Pi] + ArcSin[13/14])] + 70/123 Sqrt[7/3] Sin[1/3 (2 \[Pi] + ArcSin[(3 Sqrt[3/7])/2])] + 67/246 Sqrt[7/3] Sin[ 1/3 (4 \[Pi] + ArcSin[(3 Sqrt[3/7])/2])] + (469 Sin[ 1/3 (4 \[Pi] - ArcSin[(3 Sqrt[3])/14])])/(246 Sqrt[ 3]) - (35 Sin[1/3 ArcSin[(3 Sqrt[3])/14]])/(82 Sqrt[ 3]) + (245 Sin[ 1/3 (2 \[Pi] - ArcSin[(39 Sqrt[3])/98])])/(246 Sqrt[3]) - Sqrt[7/6 (-1 + Sqrt[13] Sin[1/3 (2 \[Pi] - ArcSin[293/(182 Sqrt[13])])])]]] Out[161]= {1365.427547, -49 + 49 #1 + #1^6 &}
NDE>For the 72-element group, a general formula is
27*x^6 + 9*t^3*(t+8)*x + t^3*(t+8)*(t^2+7*t+1) = 3*A^2 + t*(t+8)*B^2
where
A = 3*x^3 - t*(t+8)*x/2 - t^2*(t+8)/2 B = (6*x^2 + 3*t*x - (t^2+2*t)) / 2
it seems that this has a real pair of roots *iff* -8 < t < 0.
NDE
Lotsa nice ones in there: In[178]:= Factor[ 27*x^6 + 9*t^3*(t + 8)*x + t^3*(t + 8)*(t^2 + 7*t + 1) /. t -> -7] Out[178]= -343 - 3087 x + 27 x^6 In[179]:= Ssexy[%][[1]] Out[179]= {x -> Sqrt[7/3]/3 - (7^(2/3)*(1 + 3*Sqrt[21]))/ (3*(6*(378 - 140*Sqrt[21] + Sqrt[14*(41310 - 5832*Sqrt[21])]))^ (1/3)) + (7*(378 - 140*Sqrt[21] + Sqrt[14*(41310 - 5832*Sqrt[21])]))^(1/3)/ (3*6^(2/3))} In[180]:= ContinuedFraction[-%[[1, 2]], 105] Out[180]= {0, 9, 750141, 1, 1, 4, 50009, 4, 3, 4, 6, 1, 389, 1, 5, 7, 7, 2, 5, 1, 1, 1, 4, 64, 2, 1, 1, 1, 1, 1, 1, 2, 7, 1, 1, 4, 1, 1, 3, 1, 2, 1, 1, 10, 1, 5, 1, 1, 1, 1, 2, 12, 1, 32, 1, 13, 1, 47, 1, 2, 5, 1, 1, 1, 1, 1, 4, 6, 4, 2, 2, 2, 5, 1, 130, 1, 48, 1, 2, 4, 2, 3, 1, 1, 1, 2, 1, 16, 1, 1, 1, 34, 1, 1, 1, 4, 9, 1, 3, 33, 10, 1, 2, 4, 1} NDE>... and there are infinitely many solvable trinomials ax^8+bx+c, such as 4x^8+8x+13 (this must be well-known too)... My recipe for solving irreducible octics: Run and hide. Based on Tito Piezas's x^8-x^7+29x^2+29, whose solution, some of you may recall, involved solving the auxiliary septic 8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 for which he gave "v_i = 2(w^11+w^13+w^16+w^18)-2(w+w^12+w^17+w^28)-(w^2+w^5+w^24+w^27)+ (w^3+w^7+w^22+w^26)+(w^4+w^10+w^19+w^25)-(w^8+w^9+w^20+w^21) and one can set w_i = {t, t^7, t^23, t^25, t^16, t^20, t^24}, and t = exp(2Pi*I/29)." My septic solver instead returns something so unspeakably cumbersome that MinimalPolynomial can't even recover the septic. --rwg