I will talk about a new attack on the minimal ramification problem: Let G be a nontrivial finite group. The inverse Galois problem asks whether there exists a Galois extension N of the rational numbers Q with group G. A classical variant of this problem, the *minimal ramification problem*, asks to calculate m(G) -- the minimal number m(G) of prime numbers that ramify in N, where N varies over the G-extension of Q.

All previous attacks on the minimal ramification are based on number theoretical approaches, either by Galois cohomology and local global principles or by Galois representations; hence are restricted to the family of solvable groups, and certain matrix groups respectively. We develop a new machinery to bound ramifications in specializations, which is applicable to Galois extension of Q(t) one gets by rigidity methods.

Recall that the Batemen-Horn conjecture says that for non-associate irreducible integral polynomials f_1(X), ..., f_r(X) with positive leading coefficients and with no local obstructions (i.e.\ such that no p divides all values of the product f_1(n)...f_r(n), n runs over the integers) the number of 1< a< x such that f_1(a), ... , f_r(a) are all primes is about C x/log^r(x) for large x. Here C=C(f_1,..,f_r) is a positive constant. Combining the machinery we develop together with the Bateman-Horn conjecture or with the partial results toward the Bateman-Horn conjecture one gets by sieve methods we get surprising new results even for groups like S_n:

S_n is realizable over Q with a **bounded** number of ramified primes; in fact m(S_n)=1 conditionally on BH and m(S_n)<19 unconditionally.