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

TL;DR

This paper proves that certain polynomial equations have infinitely many solutions with prime inputs under specific algebraic and local conditions, confirming a conjecture for degrees 2 and 3.

Contribution

It introduces a new approach linking Schmidt's $h$-invariant to prime solutions, extending results to degrees 2 and 3.

Findings

Infinite prime-tuple solutions under local conditions

Validation of a conjecture for degrees 2 and 3

Connection between algebraic non-degeneracy and prime solutions

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 $3$.