Using Groebner Basis to eliminte variables but end with empty set.

Question

I want to infer the relationship between x and y (which is x == y) as the code (mathematica) below. However l2 gives an empty set {} when I eliminate A, B.

I wonder what's the theorem beind it?

It is because situation A==B==0 ?

Is that possible to add an assumption of A!=0 && B !=0 when computating Groebner Basis?

l2 = {
   x A - y B,
   A - B
   };
l3 = {
   4 x - 4 y,
   A - B
   };
GroebnerBasis[l2, {}, {A, B}]
GroebnerBasis[l3, {}, {A, B}]

Got answer

{}
{x - y}

Answer 1

The assumptions $A \neq 0$ and $B \neq 0$ can taken into account by adding variables Ainv, Binv for the inverses of A and B, and adding the polynomials A*Ainv - 1, B*Binv - 1 to the list of generators. Eliminating A, B, Ainv, Binv will then yield {x - y}.