Consider the identity:

$$ \begin{aligned} f_1 &= 4 (4 x + z) \cdot z^{3} \\ f &= x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4} + 4 x^{3} z + 12 x^{2} y z + 12 x y^{2} z + 4 y^{3} z + 6 x^{2} z^{2} + 12 x y z^{2} + 6 y^{2} z^{2} + 4 y z^{3} \\ f_1 + f &= (x + y + z)^{4} \end{aligned} $$

Sufficient condition $(f_1,f,f+f_1)$ to be an $abc$ triple of quality $4/(3+4/k)$ is:

$\gcd(f_1(x,y,z),f(x,y,z))$ is small (better $1$)

$f(x,y,z)$ represent $k$-powerful number so the radical is small

$|z|$ is sufficiently large so $z^3$ doesn't ruin the argument with the degrees. Probably $ |z| \ge C \max(|x|,|y|)$ will do.

This follows from the argument for abc for polynomials and there might be more conditions.

(1) and (2) appear easy. I am pretty sure $f(x,y,z)$ primitively represents infinitely many arbitrary large powers. Likely $f$ is surjective.

Experimentally the sufficient conditions are possible.

E.g. (x,y,z) $-8, 13, -31$ and $-3, 26, -10$ lead to triples $1419857, 456976, 1876833$ and $ 22000, 6561, 28561$ of qualities $1.1433$ and $ 1.2267 $ and $-15, 23, -10$ to quality $1.5679$.

For fixed $z$, $f(x,y,z)=A$ is genus $0$.

Q1 Is there a reason the sufficient conditions to happen finitely often?

Q2 How are good triples found explained?