"Source: Text/Commut/GB.text";
"Line: 2438";
"Date: Thu Sep 25 14:47:46 2025";
"Main: Fri Sep 26 14:29:19 2025";
// original file: Text/Commut/GB.text, line: 2438
// Example: H115E9 ()
print "Example: H115E9";
ei := GetEchoInput();
SetEchoInput(true);
P<x, y> := PolynomialRing(RationalField(), 2);
S := [x^2 - y, x^3 + y^2, x*y^3 - 1];       
I := IdealWithFixedBasis(S);                              
1 in I;
assert $1;
C := Coordinates(I, P!1);
C;
C[1]*S[1] + C[2]*S[2] + C[3]*S[3];
assert $1 eq 1;
P<x, y> := PolynomialRing(IntegerRing(), 2);
S := [x^2 - y, x^3 + y^2, x*y^3 - 1];       
I := IdealWithFixedBasis(S);                              
1 in I;
assert not $1;
GroebnerBasis(I);
C := Coordinates(I, P!2);
C;
C[1]*S[1] + C[2]*S[2] + C[3]*S[3];
assert $1 eq 2;
SetEchoInput(ei);
