/**/ Use R ::= QQ[x,y,z];
/**/ F := x^2*y +x*y^2 +y^2;
/**/ L := [x*y-1, y^2-1];
/**/ DivAlg(F, L);
record[quotients := [x +y, 1], remainder := x +y +1]
/**/ D := It;
/**/ D.quotients;
[x +y, 1]
/**/ D.remainder;
x +y + 1
/**/ ScalarProduct(D.quotients, L) + D.remainder = F;
true
/**/ R2 := NewFreeModule(R,2);
/**/ V := ModuleElem(R2, [x^2+y^2+z^2, x*y*z]);
/**/ L := gens(SubmoduleRows(R2, mat([[x,y], [y,z], [z,x]])));
/**/ D := DivAlg(V, L);
/**/ indent(D);
record[
quotients := [x, -z^2 +y +z, y*z -y],
remainder := [z^2, z^3 -y*z -z^2]
]
/**/ sum([D.quotients[i]*L[i] | i in 1..len(L)]) + D.remainder;
[x^2 +y^2 +z^2, x*y*z]
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