Basis of invariant polynomial system

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If you just want to compute a truncated Gröbner basis of your system (say, out to degree 7), then it's easy:

sage: R.<S0,S1,S2,S3,S4,S5> = QQ[]
sage: P.<X0,X1,X2, Y0,Y1,Y2> = Frac(R)[]
sage: P
Multivariate Polynomial Ring in X0, X1, X2, Y0, Y1, Y2 over Fraction Field of Multivariate Polynomial Ring in S0, S1, S2, S3, S4, S5 over Rational Field
sage: I = P * [
....: X0  + Y0 - S0,
....: X0 *  X1  + Y0 *  Y1   -     S1,
....: X0* ( X1^2 + 2 * X2 )+ Y0* ( Y1^2 + 2* Y2 )-   2* S2,
....: X0* ( X1^3 + 6 * X2* X1 )+ Y0 *( Y1^3 + 6* Y2 *Y1 )  -   6 *S3,
....: X0* ( X1^4 + 12* X2* X1^2 + 12* X2^2 )+ Y0 *( Y1^4 + 12* Y2* Y1^2 + 12* Y2^2 )-  24* S4,
....: X0* ( X1^5 + 20* X2* X1^3 + 60* X2^2* X1 )  + Y0 *( Y1^5 + 20 *Y2* Y1^3 + 60* Y2^2 *Y1 )  - 120* S5]
sage: I
Ideal (X0 + Y0 - S0, X0*X1 + Y0*Y1 - S1, X0*X1^2 + Y0*Y1^2 + 2*X0*X2 + 2*Y0*Y2 - 2*S2, X0*X1^3 + Y0*Y1^3 + 6*X0*X1*X2 + 6*Y0*Y1*Y2 - 6*S3, X0*X1^4 + Y0*Y1^4 + 12*X0*X1^2*X2 + 12*Y0*Y1^2*Y2 + 12*X0*X2^2 + 12*Y0*Y2^2 - 24*S4, X0*X1^5 + Y0*Y1^5 + 20*X0*X1^3*X2 + 20*Y0*Y1^3*Y2 + 60*X0*X1*X2^2 + 60*Y0*Y1*Y2^2 - 120*S5) of Multivariate Polynomial Ring in X0, X1, X2, Y0, Y1, Y2 over Fraction Field of Multivariate Polynomial Ring in S0, S1, S2, S3, S4, S5 over Rational Field
sage: %time G = I.groebner_basis(deg_bound = 7)
CPU times: user 0.17 s, sys: 0.01 s, total: 0.18 s
Wall time: 0.46 s
sage: len(G)
18

However, Note two things:

"Usual Gröbner bases" will always destroy the symmetry of your original equations. I am afraid it seems that there is no SAGBI basis in Sage. I wish it was.

Also note that the result is, in general, not a "Gröbner basis out to degree n" but a "truncated Gröbner basis". Namely, while computing the Gröbner basis, all polynomials with leading monomial of degree greater than n will simply be discarded of. But for inhomogeneous systems (like yours) it could happen that the s-polynomial of two polynomials of degree > n have a leading monomial of degree smaller than n. Hence, the truncation will lose information.

Even though you haven't told us enough data to actually see what the problem is, there is certainly no canned one-liner in Sage that does everything that you want. I'll assume that every equation is invariant under the group, otherwise you'll have to do more representation theory. Your equations also need to be of fixed total degree, otherwise you might get the wrong answer if you truncate your Groebner basis computation at some maximum degree. There is some discussion in http://trac.sagemath.org/sage_trac/ti... on that very subject.

Roughly, you probably want to do the following. First, compute the Hironaka decomposition of your invariant ring. Express the equations in terms of the primary and secondary invariants. Run the Groebner basis computation in the invariant ring.