Use R ::= QQ[x,y,z];
F := x^2y+xy^2+y^2;
L := [xy-1,y^2-1];
DivAlg(F,[xy-1,y^2-1]);
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
-------------------------------
V := Vector(x^2+y^2+z^2,xyz);
L := [Vector(x,y), Vector(y,z), Vector(z,x)];
DivAlg(V, L);
Record[Quotients = [0, -z^2, yz], Remainder = Vector(x^2 + y^2 + z^2, z^3)]
-------------------------------
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