Matthew Skerritt , University of Newcastle

Extension of PSLQ to Algebraic Integers
Given a vector $(x_1,\dots, x_n)$ of real numbers, we say that an integer relation of that vector is a vector of integers $(a_1,\dots,a_n)$ with the property that $a_1 x_1 + \dots + a_n x_n = 0$. The PSLQ algorithm will compute (to within some precision) an integer relation for a given vector of floating point numbers. PSQL is also known to work in the case of finding gaussian integer relations for an input vector of complex numbers. We discuss recent endeavours to extend PSQL to find integer relations consisting of algebraic integers from some algebraic extension field (in either the real, or complex cases). Work to date has been entirely in quadratic extension fields. We will outline the algorithm, and discuss the required modifications for handling algebraic integers, problem that have arisen, and challenges to further work.