The determinant method strikes again! (Siegel zeroes edition)

Ben Green has a nice clean writeup of the argument that there are no Siegel zeroes, found in a big OpenAI experiment a few weeks ago. Green’s writeup is itself a product of iterated back-and-forth with another OpenAI model. But it reveals something quite nice; that at the heart of the proof is the good old determinant method of Bombieri-Pila and Heath-Brown! (Except it crucially uses multiple primes at once, so I ought to mention Salberger and Walsh too, and probably some other people.) The driving idea: if you have a bunch of low-height points on a variety X, and an equally-sized bunch of polynomials on X, you can make a big old matrix by evaluating each polynomial on each point, and then you can ask this matrix what its determinant is about. Because you’ve bounded the height of the points, that determinant can’t be too archimedeanly big. On the other hand, if p is a not-too-large prime, a lot of those points reduce to the same point mod p (or even higher powers of p), which means a…