Zeroes of polynomials with prime inputs and Schmidt's $h$-invariant

Stanley Yao Xiao; Shuntaro Yamagishi

doi:10.4153/s0008414x19000026

Zeroes of polynomials with prime inputs and Schmidt's $h$-invariant

Number Theory 2019-11-13 v3

Authors: Stanley Yao Xiao , Shuntaro Yamagishi

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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 of B. Cook and \'{A}. Magyar \cite{CM} for hypersurfaces of degrees $2$ and $3$ .

Keywords

polynomial equations prime-tuple solutions Schmidt's h-invariant algebraic non-degeneracy local conditions hypersurfaces

Cite

@article{arxiv.1512.01258,
  title  = {Zeroes of polynomials with prime inputs and Schmidt's $h$-invariant},
  author = {Stanley Yao Xiao and Shuntaro Yamagishi},
  journal= {arXiv preprint arXiv:1512.01258},
  year   = {2019}
}

Comments

27 pages, revised

Zeroes of polynomials with prime inputs and Schmidt's $h$-invariant

Abstract

Keywords

Cite

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