Post #4480889

The first phase of the #SAIR / #LFMDB inverse Galois challenge https://competition.sair.foundation/competitions/igp24/overview will close in a week. One notable milestone of the competition so far: the inverse Galois conjecture has now been verified for all 25,000 of the transitive groups acting on 24 elements, by explicitly matching each such group with a degree 24 polynomial with that group as its Galois group! Prior to the competition, only 79 of these groups had been so matched. In some ways, our coverage of the degree 24 case is now slightly better than the degrees less than 24: of the 5512 eligible transitive groups in that case, all but one is known to be matched to a Galois group (the exception being the Matthieu group M_23). This does not mean that Stage 1 has been solved prematurely! We are not just tracking the Galois group of a polynomial, but also the number of real roots, increasing the number of signatures from 25,000 to 160,164. Currently, all but 5,672 of these signatures are matched to a polynomial. We are also tracking the discriminants of the polynomials submitted, and there is scope to improve them further. In Stage 2, we will reveal all the polynomials submitted, and switch to a collaborative mode in which contributors post their polynomial-finding techniques. We will then use AI to create a "superteam" who will be tasked with beating the best Stage 1 polynomials using all the ideas contributed. Stay tuned for more details!