Forms of differing degrees over number fields

TL;DR

This paper establishes an asymptotic count of integral solutions for polynomial systems over number fields, proving the Hasse principle and related conjectures for high-dimensional varieties, extending previous work to more general settings.

Contribution

It generalizes Skinner's and Browning-Heath-Brown's results by providing an asymptotic formula over arbitrary number fields using the Hardy-Littlewood circle method.

Findings

Proves an asymptotic formula for integral solutions over number fields.

Shows that smooth, geometrically integral varieties satisfy the Hasse principle and weak approximation under certain conditions.

Corrects an error in Skinner's original treatment of the singular integral.

Abstract

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_K^m$ satisfies the Hasse principle, weak approximation and the Manin-Peyre conjecture, if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner's number field version of the Hardy-Littlewood circle method. As a by-product, we point out and correct an error in Skinner's treatment of the singular integral.