ABC conjecture for multivariate polynomials?

Question

Is there an ABC conjecture for multivariate polynomials over $R[x_1,\dots,x_n]$ where $R$ is either $\Bbb Z$ or $\Bbb F_q$?

Answer 1

The Mason-Stothers Theorem also holds for polynomials over fields of prime characteristic. But one has to modify the assumption, that the polynomials $a(x),b(x),c(x)$ are non-constant by another condition, namely by the stronger assumption $(iii)$:

Theorem (Mason, Stothers): If $k$ is a field and $a(x)$, $b(x)$, and $c(x)$ are nonzero polynomials in $k[x]$ such that
(i) $a(x) + b(x) = c(x)$,
(ii) $\gcd(a(x),b(x)) = 1$,
(iii) at least one of the derivatives $a'(x)$, $b'(x)$, or $c'(x)$ is not $0$,

then we have $$ \max(\deg a(x), \deg b(x), \deg c(x)) \leq \deg({\rm rad}(a(x)b(x)c(x)) - 1. $$

The above link in the comments shows that there is also a generalised version for multivariate polynomials.