Some Toy Versions of Big Conjectures in Mathematics
Here we use the basic algebraic geometry to prove some toy versions of big conjectures in math, such as Fermat’s conjecture for polynomials and $ABC$-conjecture for polynomials.
1. Fermat’s conjecture for polynomials
Theorem 1.1. Let $f,g,h\in k[x]$ be nonconstant polynomials such that no irreducible divides all of $f,g,h$ where $k$ be an algebraically closed field of characteristic zero. If \[f^n+g^n=h^n,\] then $n\leq 2$.
Proof. Let $Z:=V_+(x^n+y^n-z^n)\subset\mathbb{P}^2$. Let $F,G,H\in k[s,t]$ be the homogenized $f,g,h$. Easy to see that the Fermat equation holds at whole $\mathbb{P}^1$, that is, $F^n+G^n=H^n$. Now consider \[\psi:\mathbb{P}^1\to Z\subset\mathbb{P}^2,[s:t]\mapsto[F(s,t):G(s,t):H(s,t)].\] This is nonconstant (since they are relatively prime), by Riemann-Hurwitz theorem we get \[0=g(\mathbb{P}^1)\geq g(Z)=\frac{(n-1)(n-2)}{2},\] hence $n\leq 2$. $\blacksquare$
2. $ABC$-conjecture for polynomials
Lemma 2.1. (Belyi’s theorem, 1979). Let $X$ be a connected nonsingular curve. Then the following are equivalent:
(i) $X$ defined over $\overline{\mathbb{Q}}$;
(ii) There exists a Belyi map $\phi:X\to\mathbb{P}^1$, that is, the branch Locus of $\phi$ is in the set $\{0,1,\infty\}$.
Theorem 2.2. (Mason-Stothers theorem). Let $f,g,h\in k[x]$ be nonconstant polynomials such that no irreducible divides all of $f,g,h$ where $k$ be an algebraically closed field of characteristic zero. If \[f(x)+g(x)=h(x),\] then \[h:=\max\{\deg f,\deg g,\deg h\}\leq \deg\mathrm{rad}(fgh)-1\] where $\mathrm{rad}(F)$ means the product of the distinct irreducible factors of $F$.
Proof. Let $\phi=\frac{g}{h}$ be a rational function of degree $h$ which induce $\phi:\mathbb{P}^1\to\mathbb{P}^1$.