ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Tue, 06 Mar 2012 05:34:42 -0600SAGBI-Grobner basis of an invariant polynomial systemhttp://ask.sagemath.org/question/8774/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](http://www.dmtcs.org/pdfpapers/dmAA0123.pdf) 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](http://www.sagenb.org/home/pub/4457/), 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)Tue, 06 Mar 2012 05:34:42 -0600http://ask.sagemath.org/question/8774/sagbi-grobner-basis-of-an-invariant-polynomial-system/