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### 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. The last polynomial is in fact invariant to the same G as the system, however after reading a paper on the topic by Nicolas M. Thiery: Computing Minimal Generating Sets of Invariant Rings of Permutation Groups with SAGBI-Grobner Basis, the last polynomial doesn't seem to be right or I'm missing something? Sorry for vagueness but I don't understand the topic 100%. However it should be simple to compute SAGBI basis of this system by hand or see just by looking at the result, that something is wrong here. 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)


### 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. The last polynomial is in fact invariant to the same G as the system, however after reading a original system. A paper on the topic by Nicolas M. Thiery: Computing Minimal Generating Sets of Invariant Rings of Permutation Groups with SAGBI-Grobner Basis, the last states that the remainder of dividing any polynomial doesn't seem to with any other should be right or 0, but I'm missing something? sure this is not the case here. Sorry in advance for vagueness but I don't understand the topic 100%. However it should be simple to compute SAGBI basis of this system by hand or see just by looking at the result, that something is wrong here. 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)


### 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. The last polynomial is in fact invariant to the same G as the original system. 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 should be 0, but I'm sure this is not the case here. here, or am I missing something? Sorry in advance for vagueness but - I don't understand the topic 100%. However it It should be simple however to compute SAGBI basis of this system by hand or see just by looking at the result, that something is might be wrong here. I suspect there is something wrong with the original set of polynomials... A link to the worksheet, if it helps!

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)


### 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. 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)

 5 retagged FrédéricC 2539 ●3 ●28 ●53

### 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)