# Basis of invariant polynomial system

I've been trying to compute a Grobner basis for a specific invariant polynomial system. It has 6 variables, 6 constants and 6 equations and is invariant to a group of cardinality 2. Various algorithms have been ran on it, including FGb/Gb through Maple and Singular through SAGE system. In both cases, the invariance was ignored and the computation of the Grobner basis failed to finish after hours (sometime days) of computation, while occupying all the memory available. Please note, I do not know what exactly the underlying algorithm was (Buchberger/F4/F5...). It is an engineering application and in practice, I would only need the first few polynomials of the Grobner basis, that is the ones with as low degree as possible. I'm an engineer not a mathematician, so my knowledge of the topic is very limited. I did however understood, that in case of invariant systems, a SAGBI basis (or invariant Grobner basis) can be computed much more efficiently. More important, the invariant Grobner basis can be computed "up to a given degree", which is exactly what I probably need. I got a hint that such algorithm might exist in SAGE. I've been searching through the SAGE documentation, but it seems I don't know what to search for and the system is huge. If anyone can point me to right direction it would be great! The problem:

```
X0 + Y0 - S0 = 0
X0 X1 + Y0 Y1 - S1 = 0
X0 ( X1^2 + 2 X2 )+ Y0 ( Y1^2 + 2 Y2 )- 2 S2 = 0
X0 ( X1^3 + 6 X2 X1 )+ Y0 ( Y1^3 + 6 Y2 Y1 ) - 6 S3 = 0
X0 ( X1^4 + 12 X2 X1^2 + 12 X2^2 )+ Y0 ( Y1^4 + 12 Y2 Y1^2 + 12 Y2^2 )- 24 S4 = 0
X0 ( X1^5 + 20 X2 X1^3 + 60 X2^2 X1 ) + Y0 ( Y1^5 + 20 Y2 Y1^3 + 60 Y2^2 Y1 ) - 120 S5 = 0
```

Where X0,X1,X2,Y0,Y1,Y2 are variables S0...S5 are constants, all are complex numbers.