Suppose that a finite set of polynomials in C[x,y,z] has a finite number of solutions (i.e. the generated ideal is 0-dimensional).

Suppose also that the Groebner basis wrt plex(x,y,z) is

[f(z), g(y,z), h(y,z), k(x,y,z)]

As well known, the system can be now easily solved: choose a root z0 of f, plug it into g and h and look for a common root (y0) etc.

The question is the following:
Is it true that for EVERY root z0 of f there exist y0, z0 such that (x0,y0,z0) satisfy the system?

In all the examples I have seen this is true, but I don't know whether this is true in general or there is a counterexample.

[This is not a pure Maple question but I know that some members here work in this area].

Thank you.

It is well known that fsolve usually increases (internally) Digits in order to obtain the desired accuracy.

But in the following example, it seems that fsolve highly exaggerates :-)   

restart;
N:=40:
Digits:=100:
F:=expand(mul(x-k,k=1..N)):
f:=evalf(F):
S:=[fsolve(f,complex)];

Error, (in fsolve) Digits cannot exceed 38654705646

Note that the bug does not appear if e.g. F:=expand(mul(x-k-I, k=1..N)):

with(plots):with(plottools):
p1:=plot( x^3,x=-1..1,thickness=20,color=red):
p2:=plot(-x^3,x=-1..1,thickness=20,color=blue):
p3:=display(rectangle([0.5, 1],[0.75,-1],color=green)):
p4:=display(rectangle([-1,-0.1],[1,0.1],color=yellow)):

display(p1,p2,p3,p4); # order = 4312

#  display(p3,p4,p1,p2); # order = 4312 (the same)

It seems that the rectangles are plotted first, in reversed order, and then the curves, in direct order.
Has someone an explanation?

There are many conventions for the Euler angles or other angles used to define o rotation of a 3d plot.
In Maple these angles are in the plot option orientation, but I think that the help page is not correct about them.
The same info appears in a worksheet (see ?rotateplot), so I am even more intrigued. [Note that many authors also switch phi and theta in spherical coordinates].

The help file says:

orientation=[theta, phi, psi]
This orientation specified by these angles is obtained by rotating the plot
1. psi about the x-axis,
2. then phi about the (transformed) z-axis, and
3. then theta about the (transformed) y-axis.
  These angles, given in degrees, are the Euler angles for the transformation matrix, using the axes specified. The angle psi is optional and is assumed to be 0 if not given. If the orientation option is not specified, the default value used is [55, 75, 0].

After some tests it seems that y and z should be switched, i.e. keeping the names (and order) for the angles ==>
1. psi about the x-axis,
2. then phi about the (transformed) y-axis, and
3. then theta about the (transformed) z-axis.
 
Am I right?

It seems to be a bug here:

restart;
with(Iterator):
M := CartesianProduct([1,2],[a,b],[c,d,e], rank=3):
n:=0:  for v in M do n:=n+1: print(n,v); end do:

              1, [2   b   c]
              2, [2   b   d]
Error, (in unknown) improper op or subscript selector

# for rank=9  ==>  [list, b, e]     ??