ag.algebraic geometry - Upper bound on greatest prime of bad reduction for a plane curve

The primes that are "bad" in your sense will divide the number $Res_x(Res_y(f,frac{partial f}{partial x}), Res_y(f,frac{partial f}{partial y}))$. (If I interpreted damiano's comment correctly).

All that is left is to bound this number. So:

Let $M := max (|a_{ij}|)$.

$parallel Res_y(f,frac{partial f}{partial x})parallel and parallel Res_y(f,frac{partial f}{partial y})parallel are < (2d)!M^{2d}$

$Rightarrow parallel Res_x(Res_y(f,frac{partial f}{partial x}), Res_y(f,frac{partial f}{partial y}))parallel < (2d^2)^{2d^2}((2d)!M^{2d})^{2d^2} ll (dM)^{4d^3+O(d^2)}$

So pick a random prime larger than this and then compute $gcd(Res_y(f,frac{partial f}{partial x}), Res_y(f,frac{partial f}{partial y}))$ in $mathbb{F}_p$. The complexity is $O(poly(d)times poly(log(M))$. Is this better than Groebner computations in $mathbb{Q}$? I have no idea...