Density of rational points on cyclic covers of \mathbb{P}^n

A Polynomial Sieve and Sums of Deligne Type

“…Remark 1. For f = T d we obtain the same result as in [Mun09][Theorem 1.1]. However, there is an issue in the proof of [Mun09][Theorem 1.1].…”

“…For f = T d we obtain the same result as in [Mun09][Theorem 1.1]. However, there is an issue in the proof of [Mun09][Theorem 1.1]. Munshi suggested the work around that we implement in Lemma 30.…”

“…These two remarks lead to the following Lemma 4 ([HB84], [Mun09]). Let P be a finite set of prime numbers p ≡ 1 mod d and P := |P|.…”

“…A fundamental role in Analytic Number Theory is played by the combination of sieve methods and bounds for sums involving algebraic functions as the additive and multiplicative characters. In this paper we present a blend of this type, introducing a generalization of the following results (i) The square sieve, or the more general power sieve of Heath-Brown and Munshi ([HB84], [Mun09]), (ii) Deligne's and Katz's estimates for additive and multiplicative characters in many variables ( [Del74], [Kat99] and [Kat02a]), and combining them to establish the following:…”

“…Munshi suggested the work around that we implement in Lemma 30. Moreover, the bound in [Mun09][Theorem 1.1] has been improved by Pierce and Heath-Brown for n ≥ 8 ([HP12][Theorem 2]).…”