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 class of having sufficient number variables compared to its degree. This sharpens previous results the first author in two aspects: range as well polynomials which can be handled. As consequence, we that there exist infinitely many primes p such \(p-1\) has prime divisor size \(\gg p^{2/5+o(1)}\) is value an incomplete norm form polynomial.