What is known as the Fermat's conjecture (or last theorem), proved by Taylor and Wiles (see The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles, 1995, Gerd Faltings, AMS notices or Fermat's Last Theorem, 2007, Darmon, Diamond, Taylor) states that the equation
$$ x^n+y^n =z^n$$
has no non-trivial integer solution ($xyz\neq 0$) for integer $n\ge 3$. The proof is notoriously difficult, and can be understood in full only by a few people. Sir Andrew J. Wiles was awarded the Abel prize in 2016.
The integers form a ring, and one may ask whether this conjecture applies in other rings. In the ring of polynomials with complex coefficients $\mathbb{C}[X]$, one may ask whether, for relatively prime polynomials $P(X)$, $Q(X)$ and $R(X)$
$$P^n(X)+Q^n(X)=R^n(X)$$
and it turns out that the above conjecture remains valid, and interestingly 5even surprisingly) only requires basic notions, and can be proved in 1-2 pages, using the Wronskian. This stems from the Mason's lemma (or Mason–Stothers theorem): if $A(X)$, $B(X)$ and $C(X)$ are non-trivial, relatively prime polynomials, and $A(X ) + B(X ) = C(X )$, then:
$$\max\{\deg (A), \deg(B), \deg(C)\} \le z(ABC) - 1,$$
where $z$ denotes the number of distinct zeroes.
Although a polynomial ring might seem more involved ($\mathbb{Z}$ is difficult, polynomials are easy), my understanding is that it inherits a notion of differentiation that is not present in integers, which in turn contrains the $\mathbb{C}[X]$'s structure so much that Fermat-Taylor-Wiles theorem becomes "almost trivial", since
$$\max\{\deg (P^n ), \deg (Q^n ), \deg (R^n )\} \le z((PQR)^n ) - 1,$$ thus
$$n\max\{\deg (P ), \deg (Q ), \deg (R )\} \le \deg (P )+ \deg (Q )+ \deg (R ) - 1.$$
A few days ago, the Abel prize was awarded gain to Robert P. Langlands, “for his visionary program connecting representation theory to number theory.”, relating number theory, algebra and harmonic analysis.
How much is thisinteger/polynomial Fermat's conjecture simplification related to the Langlands' program? More specifically:
- What are other examples of such drastic simplifications?
- Is this example really illustrative of Langlands' program