Use R ::= QQ[x,y,z];
F := 3x^2y + 5y^3  xy^5;
Coefficients(F);
[1, 3, 5]

ScalarProduct(Coefficients(F), Support(F)) = F;
TRUE

V := Vector(3x^2+y, x5z^3);
Coefficients(V);
[5, 3, 1, 1]

ScalarProduct(Coefficients(V), Support(V)) = V;
TRUE

Coefficients(x^3z+xy+xz+y+2z, x);
[z, 0, y + z, y + 2z]

F := (1 + 2x + 3y^4 + 5z^6)^7;
Skeleton := [1, x^3, y^12, z^19, x^2*y^8*z^12];
Coefficients(F, Skeleton);
[1, 280, 945, 0, 567000]

