The paper "Using SAGBI bases to compute invariants" by Stillman and Tsai (1990) describes algorithms for computing Sagbi bases of subrings contained in quotient rings. The following code demonstrates Example 2 from that paper (in the case $N=4$.)
N = 4;
gndR = QQ[(a,b,c,d)|(u_1..u_N)|(v_1..v_N), MonomialOrder => Lex];
I = ideal(a*b - b*c - 1);
quot = gndR/I;
U = (vars quot)_{4..(N+3)}
V = (vars quot)_{(N+4)..(2*N+3)}
G = flatten for i from 0 to N-1 list(
{a*(U_(0,i)) + b*(V_(0,i)), c*(U_(0,i)) + d*(V_(0,i))}
);
sag = sagbi G
ans = matrix {{c*u_4+d*v_4, c*u_3+d*v_3, c*u_2+d*v_2, c*u_1+d*v_1, a*u_4+b*v_4, a*u_3+b*v_3, a*u_2+b*v_2,
a*u_1+b*v_1, a*d*u_3*v_4-a*d*u_4*v_3-b*c*u_3*v_4+b*c*u_4*v_3,
a*d*u_2*v_4-a*d*u_4*v_2-b*c*u_2*v_4+b*c*u_4*v_2, a*d*u_2*v_3-a*d*u_3*v_2-b*c*u_2*v_3+b*c*u_3*v_2,
a*d*u_1*v_4-a*d*u_4*v_1-b*c*u_1*v_4+b*c*u_4*v_1, a*d*u_1*v_3-a*d*u_3*v_1-b*c*u_1*v_3+b*c*u_3*v_1,
a*d*u_1*v_2-a*d*u_2*v_1-b*c*u_1*v_2+b*c*u_2*v_1}}
assert (gens sag == ans);
|
In general, when a finite SAGBI basis does happen to exist, the algorithm *should* be able to calculate it correctly given enough time. However, there are some peculiarities in the case of quotient rings, such as ``false termination" Example 1 from the same paper:
i1 : gndR = QQ[x,y, MonomialOrder => Lex]; |
i2 : I = ideal(x^2 - x*y); o2 : Ideal of gndR |
i3 : Q = gndR/I; |
i4 : subR = sagbi subring {x};
|
i5 : gens subR
o5 = | x |
1 1
o5 : Matrix Q <--- Q
|
Although the initial algebra in this example is infinitely generated, new generators are not generated as expected from S-pairs.