# General solution to the U(1) anomaly equations

## TL;DR

This paper provides a complete parametrization of solutions to the U(1) anomaly cancellation equations, which are cubic Diophantine equations, thereby offering a general method to solve these equations for any number of Weyl fermions.

## Contribution

It introduces a parametrization of the U(1) anomaly equations as a cubic Diophantine equation and proves its generality, advancing the understanding of anomaly cancellation conditions.

## Key findings

- Derived a parametrization of U(1) charges in terms of fewer integers.
- Proved the parametrization is the most general solution.
- Solved the cubic Diophantine equation for anomaly cancellation.

## Abstract

The anomaly cancellation equations for the $U(1)$ gauge group can be written as a cubic equation in $n-1$ integer variables, where $n$ is the number of Weyl fermions carrying the $U(1)$ charge. We solve this Diophantine cubic equation by providing a parametrization of the charges in terms of $n-2$ integers, and prove that this is the most general solution.