Formalization of Polynomial ABC and FLT

• Jineon Baek (University of Michigan, Ann Arbor -- [email protected])
• Seewoo Lee (University of California, Berkeley -- [email protected])

This is a formalization of the proof of ABC theorem on polynomials (Mason-Stothers Theorem) and their corollaries (nonsolvability of Fermat-Catalan equation and FLT for polynomials, Davenport's theorem) in Lean 4. More precisely, we formalized the proofs of the following theorems:

Theorem (Mason-Stothers, Polynomial ABC) Let $k$ be a field. If $a, b, c \in k[X]$ are nonzero and $a + b + c = 0$ and they are coprime to each other, then either $\text{max}(\text{deg } a, \text{deg }b, \text{deg }c) < \text{deg} (\text{rad } a b c)$ or all $a', b', c'$ are zero.

Corollary (Nonsolvability of Fermat-Catalan equation) Let $k$ be a field and $p, q, r \geq 1$ be integers satisfying $1/p + 1/q + 1/r \leq 1$ and not divisible by the characteristic of $k$. Let $u, v, w$ be units in $k[X]$. If $ua^p + vb^q + wc^r = 0$ for some nonzero polynomials $a, b, c \in k[X]$, then $a, b, c$ are all constant polynomials.

Corollary (Polynomial FLT) If $n \geq 3$, the characteristic of $k$ does not divide $n$ (this holds when characteristic is equal to zero), $a^n+b^n=c^n$ in $k[X]$, and $a, b, c$ are nonzero all coprime to each other, then $a, b, c$ are constant polynomials.

Corollary Let $k$ be a field of characteristic $\neq 2, 3$. Then the elliptic curve defined by the Weierstrass equation $y^2 = x^3 + 1$ is not parametrizable by rational functions in $k(t)$. In other words, there does not exist $f(t), g(t) \in k(t)$ such that $g(t)^2 = f(t)^3 + 1$.

Corollary (Davenport's theorem) Let $k$ be a field (not necessarily has characteristic zero) and $f, g \in k[X]$ be coprime polynomials with nonzero deriviatives. Then we have $\deg (f) + 2 \le 2 \deg (f^3 - g^2)$.

The proof is based on the online note by Franz Lemmermeyer, which is a slight variation of Noah Snyder's proof (An Alternate Proof of Mason's Theorem, Elem. Math. 55 (2000) 93--94). See proof_sketch.md for details.

Installation

After you install Lean 4 properly (see here for details), run the following commands (or their equivalents):

# clone the repository
git clone https://github.com/seewoo5/lean-poly-abc.git
cd lean-poly-abc
# get mathlib4 cache
lake exe cache get

Gitpod

Using Gitpod, you can compile the codes on your browser. Sign up to Gitpod and use the following URL:

gitpod.io/#https://github.com/seewoo5/lean-poly-abc

Paper

arXiv version of the paper can be found here: link

Acknowledgement

Thanks to Kevin Buzzard (@kbuzzard) for recommending this project, and also Thomas Browning (@tb65536) for helping us to get start with and answer many questions.