Suppose you want to write the following expression  Expr  in powers of  x-1 . You can do it like this:

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Since the curve  f(x)  is symmetrical about the Oy axis, the circle must also be symmetrical about the Oy axis and touch the lower branch in the point  (0,-1) . So the upper semicircle will be  y=sqrt(R^2-x^2)+R-1 . In general, we have 5 points of intersection. If we impose the condition  R^2-2*R = 0 , then we have 3 points of intersection:

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Download problem_new.mw

restart;
i := [seq(2*i-1, i = 1 .. 10)];
A[m]:=[seq((x/a)^(i+1)*(1-x/a)^2, i in i)];

I think OP meant the intersection not of the surfaces of these tori, but of the solids themselves. Here is a way:

restart;
with(plots):
ce := (p, q, r, a, b) ->  ((x-p)^2 + (y-q)^2 + (z-r)^2 + a^2 - b^2)^2 - 4*a^2*((x-p)^2 + (y-q)^2):
T1 := ce(1, 1, 1, 2, 1):
T2 := ce(1, 6, 1, 2, 1):
implicitplot3d(max(T1,T2), x=-1..3, y=0..4.5, z=-1..2, style=surface, color="Red", grid=[50, 50, 50], scaling=constrained, axes=normal, orientation=[15,80]);

Example:

RandomTools:-Generate(choose({x->x^2,x->x^3,x->x^4}));

You are actually solving the inequality with a parameter ( t1 is a parameter). I don't know about the latest versions, but Maple 2018 does not support this, an error message appears:

restart;
solve((4*t1+4*sqrt(t1^2-4*t2))>0,t2, allsolutions, parametric=true);

              Error, (in solve) invalid input: SolveTools:-SemiAlgebraic expects its 1st argument, osys, to be of      type ({list, set})({ratpoly(rational), ratpoly(rational) = ratpoly(rational), ratpoly(rational) <> ratpoly(rational), ratpoly(rational) <= ratpoly(rational), ratpoly(rational) < ratpoly(rational)}), but received {0 < 4*t1+4*(t1^2-4*t2)^(1/2)}

The problem is solved with good accuracy by reducing to a system of 4 equations with 4 unknowns:

Download Square.mw

I do not see any problems in this to waste my time on such manipulations. If anyone likes one form, then he can easily switch to it using the  convert  command.

Examples:

restart;
convert(sec(x), sincos); 
convert(csc(x), sincos);
convert(tan(x), sincos);
convert(sin(x)/cos(x), tan);

P:=a->plots:-display(
	tubeplot([seq](L(s)), s=-2..2.5, radius=0.08, color="Red"),   # the line L
	tubeplot([seq](C(s)), s=-4*Pi..4*Pi, radius=0.08, color="Green", numpoints=200), # the curve C
	plot3d(S(s,t), s=-4*Pi..4*Pi, t=0..a, color="Gold", grid=[80,25]),
	scaling=constrained, style=patch, lightmodel=light4,
	orientation=[-120,75,0], labels=[x,y,z], axes=framed):

plots:-animate(P,[a], a=0..2*Pi, frames=90);

The answer has been edited.
 

Download SurfaceMP_new1.mw

 

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restart;
with(algcurves):

f:=2*z^6 + z^7/2 - (5*z^11)/4 + 4*z^22 + (29*z^34)/10 - z^40 - (13*z^43)/2 + w^38*(z^2 - z^7/4) + 
 w^49*(-z^9 + z^13/4 + 2*z^14) + w^34*((7*z^14)/3 - (3*z^18)/2) + w^47*(z^10/3 + (7*z^11)/4 + (8*z^21)/5) + 
 w^24*(4*z^8 + (4*z^25)/5 - (3*z^27)/2) + w^9*((-6*z^2)/5 - z^6/2 + (7*z^31)/3) + 
 w^16*((7*z^21)/3 + (4*z^27)/5 + (4*z^32)/3) + w^18*(-6*z^14 - 2*z^31 - z^33) + w^3*(2*z^17 + (7*z^34)/2) + 
 w^16*((-3*z^5)/4 - 2*z^36 + z^39/3) + w^50*(-1/3*z^23 - (7*z^40)/2 + z^42) + w^4*((-3*z^30)/2 + (4*z^38)/3 + (8*z^42)/5) + 
 w^33*(-3*z^4 + (8*z^22)/3 - (8*z^43)/5) + w^16*(-1/4*z^26 - (3*z^41)/4 - z^43) + w^48*((2*z^2)/3 + 6*z^26 + (3*z^43)/5) + 
 w^49*(2*z^18 + z^36 - 2*z^44) + w^10*((-2*z^11)/5 - (3*z^26)/2 + z^45) + w^40*(-1/2*z^20 - z^29 + z^46) + 
 w^36*(-4 + 8*z^13 - (7*z^47)/4) + w^14*((7*z^24)/5 - 6*z^32 - 6*z^49) + w^22*(-2*z^27 - (8*z^50)/3) + 
 w^2*((3*z^10)/5 + (7*z^24)/4 - z^50/4);

genus(f,z,w);

                                                         2268

restart;
L:=[k$1,y$23,f$25];
L0:=sort(ListTools:-Collect(L), key=(t->t[2]));
map(t->t[1], L0); 
map(t->t[2], L0);

Yes, this is indeed an ellipse, most of which is below the Ox axis and it is strongly elongated along this axis. To see it, you need to lengthen the axes and use the scaling=constrained option to correctly show its shape:

plots:-implicitplot(1/1350000000000000000*x^2-1/14580000000000000000000000000000000*x*y+1/5400000000000000*y^2-1/2250000000*x+173/5400000000*y-1=0, x=-10^10..10^11, y=-3*10^8..3*10^7, gridrefine=3, scaling=constrained, size=[1000,100]);