Date: Sun, 24 May 2015 03:39:38 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] algebraic fcn simplification Content-Type: text/plain; charset=UTF-8

I just spent hours simplifying (64 t^2 - 16 Sqrt[-32 t^2 + t^2/Sqrt[1 - t] + 31 Sqrt[1 - t] t^2 + 32 t^3 - 17 Sqrt[1 - t] t^3 + 10 t^4 + Sqrt[1 - t] t^4])/(128 t^2)

to 1/2 + (4 - 5 t)/(8 (1 - t)^(1/4) t) - ((1 - t)^(1/4) (4 + t))/(8 t) using Corey's denester on numeric special cases. How should I have done this?

Plotting the difference shows they only agree when arg t lies within ?/4 of the real axis. Elsewhere it seems to be -1. Mma gives 0 for Limit t->0, but Plot3D rather consistently spikes at the origin. I don't know how to elicit that spike value except by plotting. Bug? --rwg

I think that is only a kind of numerical artefact of the plotting routine, which may not cancel out expressions in t=0 (or t=1) If I call a resp. b your expressions, then using Maple they are both roots of the polynomial P(y) = t^4 - 256*(-1+t)*(3*t-4)*(t-4)*y +256*(-1+t)*(19*t^2-16*t+16)*y^2 - 8192*t^2*(-1+t)*y^3 + 4096*t^2*(-1+t)*y^4 The limit for those roots in t=0 are -infinity*I, infinity*I, 0, 1 and one sees that a and b must coincide.