Weird behaviour with GroebnerBasis in v7

Question

I came across some weird behaviour when using GroebnerBasis. In m1 below, I used a Greek letter as my variable and in m2, I used a Latin letter. Both of them have no rules associated with them. Why do I get vastly different answers depending on what variable I choose?

Image:

Copyable code:

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:

As pointed out by belisarius, my usage of GroebnerBasis is not entirely correct as it requires a polynomial input, whereas mine is not. This error, introduced by a copy-pasta, went unnoticed until now, as I was getting the answer that I expected when I followed through with the rest of my code using m1 from above. However, I'm not fully convinced that it is an unreasonable usage. Consider the example below:

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]}}

which is correct. So my interpretation is that it is OK to use GroebnerBasis in such cases, but I'm not all too familiar with the deep theory behind it, so I could be completely wrong here.

_{P.S. I heard that if you mention GroebnerBasis three times in your post, Daniel Lichtblau will answer your question :)}

Answer 1

The bug that was shown by these examples will be fixed in version 9. Offhand I do not know how to evade it in versions 8 and prior. If I recall correctly it was caused by an intermediate numeric overflow in some code that was checking whether a symbolic polynomial coefficient might be zero.

For some purposes it might be suitable to specify more variables and possibly a non-default term order. Also clearing denominators can be helpful at least in cases where that is a valid thing to do. That said, I do not know if these tactics would help in this example.

I'll look some more at this code but probably not in the near future.

Daniel Lichtblau

Answer 2

This may be related to the fact that Mathematica does not try all variable orders in functions like Simplify. Here is an example:

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)

Adam Strzebonski explained that:

...one can try FullSimplify with all possible orderings of chosen variables. Of course, this multiplies the computation time by Factorial[Length[variables]]...