Mason–Stothers theorem

Remarks

When $K$ is algebraically closed, $\deg \operatorname{rad}(abc)$ is the number of distinct roots of $abc$. If $K$ has characteristic $0$, the hypothesis on derivatives just says that $a$, $b$, $c$ are not all constant. A short proof, due to Noah Snyder, studies the Wronskian $ab' - a'b$: it is nonzero, it is divisible by $abc/\operatorname{rad}(abc)$, and its degree is less than $\deg a + \deg b$. Example: over $\Q$, take $a = t^2$, $b = 2t + 1$ and $c = (t + 1)^2$; then $\operatorname{rad}(abc) = t(2t + 1)(t + 1)$ has degree $3$, and the bound $2 \le 3 - 1$ is sharp. The derivative hypothesis is necessary: over a field of characteristic $p > 0$, take $a = t^p$, $b = 1$, $c = (t + 1)^p$; then $a' = b' = c' = 0$, the radical has degree $2$, and $\deg c = p > 1$. A consequence: for $n \ge 3$ there are no pairwise coprime polynomials $x, y, z$ over $\C$, not all constant, with $x^n + y^n = z^n$.

Going deeper

For a student

The theorem is the polynomial form of the abc conjecture, with the degree playing the role of the logarithm of the size and the radical counting distinct prime factors. In contrast with the integer case, it has a short and complete proof by differentiation. It gives quick proofs of Fermat's Last Theorem for polynomials and of related results on equations such as $x^2 - y^3 =$ a polynomial.