# SAGBI-Grobner basis of an invariant polynomial system

Hi all!

I've been looking into the SAGBI-Grobner basis and I gave it a test drive. My very simple multivariate system is invariant and I want to find its SAGBI basis via the Singular library. I'm pretty sure the code below produces *wrong* result for some reason (I'm by no means saying there is a bug in the Singular lib).

The last polynomial is in fact *invariant* to the same G as the original system, the rest of the polynomials are identical to the original ones. A paper on the topic by Nicolas M. Thiery: Computing Minimal Generating Sets of Invariant Rings of Permutation Groups with SAGBI-Grobner Basis states that the remainder of dividing any polynomial with any other in the SAGBI basis should be 0, but I'm sure this is not the case here, or am I missing something? Sorry in advance for vagueness - I don't understand the topic 100%. It should be simple however to compute SAGBI basis of this system by hand or see just by looking at the result, that something might be wrong here. I suspect there is something wrong with the original set of polynomials... A link to the worksheet, if it helps!

Thank you in advance,

Sash

```
R1.<S0,S1,S2,S3> = QQ[]
P1.<X0,X1, Y0,Y1> = Frac(R1)[]
I1 = P * [
X0 + Y0 - S0,
X0 * X1 + Y0 * Y1 - S1,
X0* ( X1^2 )+ Y0 *( Y1^2 ) - 2* S2]
PI = singular(I1)
singular.LIB("sagbi.lib")
PI.sagbi()
------------------------------------------------
X0+Y0+(-S0),
X0*X1+Y0*Y1+(-S1),
X0*X1^2+Y0*Y1^2+(-2*S2),
X0*X1^2*Y0-2*X0*X1*Y0*Y1+X0*Y0*Y1^2+(-S0)*X0*X1^2+(-S0)*Y0*Y1^2+(2*S1)*X0*X1+(2*S1)*Y0*Y1+(-2*S2)*X0+(-2*S2)*Y0+(2*S0*S2-S1^2)
```