Zeroes of Polynomials With Prime Inputs and Schmidt's $h$-invariant. (7th June 2020)

• Record Type: Journal Article
• Authors: Xiao, Stanley Yao; Yamagishi, Shuntaro
• Is Part Of: Canadian journal of mathematics. Volume 72:Number 3(2020)
• Page Start: 805
• Page End: 833
• Publication Date: 2020-06-07
• Subjects: 11D72, -- 11P32; Circle method, -- h-invariant, -- Hardy–Littlewood, -- prime numbers; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X19000026
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions, assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$ -invariant introduced by W. M. Schmidt. Our results prove a conjecture by B. Cook and Á. Magyar for hypersurfaces of degree 3.