Can I use groebner basis to solve polynomial equation of this kind?

Question

I want to solve a polynomial equation of the following kind.

$a (x+y^2-3z) + b (x^2 + 3y^3) = 0$

I want to compute integer coefficients a, b s.t. the above equation is satisfied for all x,y,z. In reality, my equation has 1000s of variables and is a large equation.

I converted it to solving a set of linear equations by equating coefficient of every monomial of x,y,z to 0, and then solved for a,b. But this is taking lot of time for my problem. Is there a more efficient way to do this? Can I somehow use groebner basis for my problem?

Answer 1

Well, if you have a single polynomial equation, the Groebner basis approach will not be helpful since the Groebner basis will consist of the single polynomial involved in the equation.