# Uniform bounds for rational points on complete intersections of two quadric surfaces

## TL;DR

This paper establishes uniform upper bounds on the number of rational points of bounded height on smooth complete intersections of two quadrics in projective three-space, using a combination of determinant methods and descent techniques.

## Contribution

It introduces a novel approach combining determinant methods with descent to derive uniform bounds for rational points on these intersections.

## Key findings

- Established explicit uniform bounds for rational points.
- Demonstrated effectiveness of combined determinant and descent methods.
- Applicable to non-singular complete intersections of two quadrics.

## Abstract

We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent arguments.