Test post: Zoomed out philosophy

This is mainly a test post to check whether the LaTeX support works here and to discuss briefly what may in broadest terms be thought of as the philosophy of Mochizuki’s proof.

In polynomials, there is a very easy proof of the ABC conjecture (more correctly, the Mason-Stothers theorem) due to Noah Snyder. We give that proof here.  If

a(x)+b(x)=c(x)  with a(x),b(x),c(x) relatively prime polynomials over the complex numbers then we also have a^{\prime}(x)+b^{\prime}(x)=c^{\prime}(x). From this we see that

a(x) b'(x)-a'(x) b(x)=c(x) b'(x)-c'(x) b(x)

Another way of saying that is that the wronskian W(a,b) of a and b is the same as W(c,b). We let R be the product of all factors which are common to either a and a^{\prime} or b and b^{\prime} or c and c^{\prime}. Then R is the quotient of abc by the radical. If for instance a has degree n and b has degree n-k and c has degree n then the Wronskian W(a,b)=W(b,c) has degree at most  2n-k-1 so the radical has degree at least n+1. All other cases follow in the same way.

What we learn from this is that being able to take the derivative helps to prove the abc conjecture. In the integer case, we should be taking the derivative in the direction of choices of primes. It is probably for this reason that Mochizuki wants to monkey around with the definition of schemes and introduce so many objects. He hopes to find a way to make sense of this differentiation process.

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