Galois Theory

Naming the Galois group of a degree-24 integer polynomial ordinarily demands resolvent computations inside a computer algebra system. This paper presents a statistical alternative that needs only polynomial factorisation modulo small primes, and evaluates it against 576,682 polynomials whose groups the SAIR IGP24 evaluation server computed independently in Magma. The method reads the multiset of Frobenius cycle types at 60 primes as a sample from the group’s own cycle-type distribution, which the Chebotarev density theorem licenses, then scores that sample against empirical profiles of candidate groups drawn by product-replacement sampling. Within its domain the classifier reaches 69.0% top-1 and 88.7% top-3 accuracy at 53 ms per polynomial.

There is a question in mathematics that has been open since the 1800s, and for one slice of it you can now attack it by search. I built a factory that produced degree-24 polynomials, submitted ten thousand scoreable pairs to a competition organised with the LMFDB and Terence Tao, and finished 54th of 256 with a score of 2.36. The interesting part is not the rank. It is that I can prove, with numbers, exactly why the score was small, and the answer says something about how these searches should be run.