Solid people try showstat(convexhull).

@Markiyan Hirnyk 

In convexhull the coordinates are converted to floats, so the irrationals are not treated properly.

My favorite  "Next Number" Puzzle:

1  2  4   8  16  ?

"Correct" answer: 31

Proof:

with(CurveFitting):
PolynomialInterpolation([[0,1],[1,2],[2,4],[3,8],[4,16]], x):
eval(%,x=5);
      31

@taro 

In an ideal world, simplify (and a fortiori simplify(...,size)  which is now the default) should be enough.

@acer 

If f is a multivalued mapping, in many situations Maple returns z0 as a solution for f(z)=0, if for some branch f(z0)=0.

I don't think that this behavior is going to change soon. Of course, sqrt is not multivalued (being defined in the help page as principal branch) but it is treated as such.
Note that your equation, for x <>0 reduces to csgn(x)=1 but
solve(csgn(x)=1, x) ;

gives

Warning, solutions may have been lost
                      RootOf(csgn(_Z) - 1)

The correct solution of solve should be

   0,  RootOf(csgn(_Z) - 1)

but I am skeptical about such an answer.

Edit. The Maple answer for your second example is correct if the above "some branch" approach is adopted 

@Markiyan Hirnyk 

In computing the Groebner basis the monomial order is essential in the elimination process.

@Markiyan Hirnyk 

You have the wrong order. Why did you change my correct one?

BTW, Kitonum's solution is equivalent to my "A more direct ...".

@Markiyan Hirnyk 

Groebner:-Basis([x -(u-v)^2, y -  u^2+3*v^2, z - (1/2)*v*(u-2*v)],plex(u,v,x,y,z));
     [x-y+4*z, v^4-v^2*y+8*v^2*z+4*z^2, v^3+2*u*z-v*y+4*v*z, u*v-2*v^2-2*z, u^2-3*v^2-y]

@Markiyan Hirnyk 

It is the standard Brioschi formula for the curvature: https://en.wikipedia.org/wiki/Gaussian_curvature
(it is not really "code" needing comments).

@tomleslie 

Yes, Maxima is an open source branch of the old Macsyma.

You have joined the 5 matrices into a large 2x10 matrix and computed its singular values.
For me it's a nonsense.
@Adam Ledger

@Adam Ledger 

You should use proper mathematical terms if you want to be understood.

In this case you are thinking probably at the equivalence:
A ~ B  <==> there exists a real t <> 0 such that A = tB

and want a set of representatives. You should be aware that there are many such equivalences. The one related to the echelon form is rank(A)=rank(B), and in your case (set) they are the same.

@Pascal4QM 

Yes, I know this. But my point is that the commutator does not help here.

It is a RingTheory (or AlgebraTheory) package which is missing in Maple (preferably with simplify/siderelations). GroupTheory works pretty well.

@Pascal4QM 

Why did you use the commuator? It seems that anyway the formula must be proved by hand (via induction it's easy).

As it seems that you work with this package, and knowing that several Maple basic operators are redefined (`*` included), how can the user know whether a command outside the package (such as solve)  is reliable in the new context? Are these situations documented, or it's only "common sense"?

@alecjacobson 

The computations are simple; they can be easily included in a procedure:

Int( F(x,y[0],y[1],`...`,y[n] ), x=a..b);   # --> min, 

y[k] = Diff(y[0],[x$k]);                         # y[0] is the unknown function

y[k](a)=A[k], y[k](b)=B[k], k=0..n-1;   # boundary conditions

Sum( (-1)^k * Diff( Diff(F,y[k]), [x$k]), k=0..n ) = 0;
# E-L equation; order is 2*n, equal to the number of boundary conditions