Extending the PSLQ Algorithm to Algebraic Integer Relations

TL;DR

This paper extends the PSLQ algorithm to find algebraic integer relations in quadratic extension fields, broadening its applicability beyond real and Gaussian integers.

Contribution

The paper introduces modifications to PSLQ for algebraic integers in quadratic fields, including algorithm outline, handling challenges, and experimental validation.

Findings

Successfully extended PSLQ to quadratic extension fields

Identified key challenges in algebraic integer relation detection

Provided experimental results demonstrating the extended algorithm's effectiveness

Abstract

The PSLQ algorithm computes integer relations for real numbers and Gaussian integer relations for complex numbers. We endeavour to extend PSLQ to find integer relations consisting of algebraic integers from some quadratic extension fields (in both the real and complex cases). We outline the algorithm, discuss the required modifications for handling algebraic integers, problems that have arisen, experimental results, and challenges to further work.