Large sieve estimate for multivariate polynomial moduli and applications

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri–Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens previous results of the first author in two aspects: the range of the moduli as well as the class of polynomials which can be handled. As a consequence, we deduce that there exist infinitely many primes p such that p- 1 has a prime divisor of size ≫ p2/5+o(1) that is the value of an incomplete norm form polynomial.