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Strange behavior with GroebnerBasis in v7

When using, GroebnerBasisI came across some strange behavior. In the m1below, I used the Greek letter as my variable, and in m2I used the Latin letter. Both of them have no rules associated with them. Why do I get completely different answers depending on which variable I choose?

Picture:

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The code to be copied:

Clear["Global`*"]
g = Module[{x}, 
    x /. Solve[
      z - x (1 - b - 
           b x ( (a (3 - 2 a (1 + x)))/(1 - 3 a x + 2 a^2 x^2))) == 0,
       x]][[3]];
m1 = First@GroebnerBasis[\[Kappa] - g, z]
m2 = First@GroebnerBasis[k - g, z]

EDIT:

, GroebnerBasis , , - . , -, , , , , m1 . , . :

x = (-b+Sqrt[b^2-4 a c])/2a;
p = First@GroebnerBasis[k - x,{a,b,c}]; (*get relation or cover for Riemann surface*)
q = First@GroebnerBasis[{D[p,k] == 0, p == 0},{a,b,c},k,
    MonomialOrder -> EliminationOrder]; 

Solve[q==0, b] (*get condition on b for double root or branch point*) 

{{b -> -2 Sqrt[a] Sqrt[c]}, {b -> 2 Sqrt[a] Sqrt[c]}}

. , , GroebnerBasis, , .

< > P.S. , GroebnerBasis , :)

+5
2

, , ​​ 9. , 8 . , , , .

, , . , , , . , , .

, , , .

+4

, Mathematica , Simplify. :

ClearAll[a, b, c]
expr = (c^4 b^2)/(c^4 b^2 + a^4 b^2 + c^2 a^2 (1 - 2 b^2));
Simplify[expr]
Simplify[expr /. {a -> b, b -> a}]

   (b^2 c^4)/(a^4 b^2 + a^2 (1 - 2 b^2) c^2 + b^2 c^4)
   (a^2 c^4)/(b^2 c^2 + a^2 (b^2 - c^2)^2)

, :

... FullSimplify . , [ []]...

+3

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