Mason-Stothers theorem

Mason’s theorem is often described as the polynomial case of the (currently unproven) ABC conjecture.

Theorem 1 (Mason-Stothers).

Let $f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{,}g\mathit{}\mathrm{(}z\mathrm{)}\mathrm{,}h\mathit{}\mathrm{(}z\mathrm{)}\mathrm{\in }\mathrm{C}\mathit{}\mathrm{[}z\mathrm{]}$ be such that $f\mathit{}\mathrm{(}z\mathrm{)}\mathrm{+}g\mathit{}\mathrm{(}z\mathrm{)}\mathrm{=}h\mathit{}\mathrm{(}z\mathrm{)}$ for all $z$, and such that $f$, $g$, and $h$ are pair-wise relatively prime. Denote the number of distinct roots of the product $f\mathit{}g\mathit{}h\mathit{}\mathrm{(}z\mathrm{)}$ by $N$. Then

$\max \mathrm{deg}\{f,g,h\}+1\le N.$