@vv  Very well, +. Now we know that the polynomial interpretation also has solutions. (It is clear that a finite set of solutions of a polynomial system corresponds to a countable set of solutions of the original system.) Wonder how long it will take to find these solutions...

@C_R 
In my opinion, waiting so long is already a lot of work. Yes, I have long had suspicions about the real possibilities of solving polynomial systems on a theoretical basis. 
By the way, Draghilev's method allows us to draw conclusions about the solution of the original system itself, we can even say, to obtain its complete solution: for example, you can solve 4 of 5 equations in turn, tracking the change in the sign of the remaining equation.
In any case, I'm sure this is a very good example for Maple.

@janhardo 
maybe this is a call for everyone to use the search function on the forum more actively.

There was one activist here (Markiyan Hirnyk), so in order to protect myself from his nastiness, I turned to the community for help. But now he visits other forums.
(For some time now, I have also formally been a moderator, and, just in case, I will say right away that I have nothing to do with deleting non-advertising posts, that is, I have never deleted your posts.)

@Carl Love   I read both your messages in the thread with great pleasure. It seems to me that the online translator conveys the level of your mastery of the word.

@vv  No, it's not compact, it's professional☺.

@vv  

Thank you for your participation. Yes, it can be proved both geometrically and analytically.
This example was made specifically because of the animation. (I really like moving pictures.)

@C_R  
Well, you can try to evaluate the quality of the translation on one text from the MapleSoft application center. By the way, there is also a description of the Draghilev method. Translation from Russian to English online Google translator.

@C_R 
In other words, you don't know about the existence of online translators?

@sija 
 

I will show you my example with the rolling of the original ellipsoid
x1^2 + 5*x2^2 + 4*x3^2 - 0.25 = 0
on the transcendental surface
x3 = - 0.25*x1^2*sin(x1) - 0.25*sin(1.5*x2).

Of course, the greater the accuracy, the more time is required. To get animation on an old computer, we choose the optimal option. The equation of the ellipsoid at each point is printed on the graph.
for_an_ellipsoid.mw

(I have many different examples on this topic on the forum. I don't provide text for every example, because it's too cumbersome, but I try to show the basic program.)

@C_R  
In this case sqrt(b[5] ^2+ b[6]^2) is for the working point to move uniformly along the circle and also to make it easier to get the integration interval. But this has nothing to do with Draghilev's method. If you want, I can give you a list of literature in Russian that preceded the method, well, the method itself in Russian.
The author died long ago, and I do this as a hobby. I am not a professional. But I repeat, I will answer any questions about the Method.

@sija  I don't know, it seems to me that 2d examples are not very different from 3d ones. I would advise to immediately work on rolling the surface, or rather, simultaneously with the curve.

@sija 
Very good, +.
I really like pictures like this. And I myself sometimes have fun this way.   
Are there any plans for examples of rolling in 3D and plans for rolling of surfaces too?

@C_R  

I didn't see your message because there was no request to me. I came across this place completely by accident. Click "Reply" and then I will know because the flag in the upper right corner of the page will turn red for me.
b[i] is a way to make Maple do natural parameterization when I use internal procedures. Such actions are much easier to perform if you write the code for solving the ODE yourself.

Yes, that's right: we get successive solutions and do this based on solving the Cauchy problem. That is, we solve a system of nonlinear equations using a numerical solution of the ODE.
In this case, it is the Draghilev method.

@vv  OK, thank you.