On forms in prime variables

TL;DR

This paper proves the existence of infinitely many prime solutions to systems of homogeneous polynomial equations under certain non-singularity and local solubility conditions, extending prime solutions theory to higher-degree forms.

Contribution

It establishes conditions for prime solutions in systems of homogeneous polynomials of degree at least 2, generalizing previous results to higher degrees and multiple forms.

Findings

Infinitely many prime solutions exist under specified conditions.

Non-singular solutions over p-adic units and real cube imply prime solutions.

Results apply to systems with degree d ≥ 2 and multiple forms.

Abstract

Let $F_1,\ldots,F_R$ be homogeneous polynomials of degree $d\ge 2$ with integer coefficients in $n$ variables, and let $\mathbf{F}=(F_1,\ldots,F_R)$. Suppose that $F_1,\ldots,F_R$ is a non-singular system and $n\ge 4^{d+2}d^2R^5$. We prove that there are infinitely many solutions to $\mathbf{F}(\mathbf{x})=\mathbf{0}$ in prime coordinates if (i) $\mathbf{F}(\mathbf{x})=\mathbf{0}$ has a non-singular solution over the $p$-adic units $\U_p$ for all prime numbers $p$, and (ii) $\mathbf{F}(\mathbf{x})=\mathbf{0}$ has a non-singular solution in the open cube $(0,1)^n$.