Integer relations among algebraic numbers

Abstract

A vector m=(m ₁,...,m _n ) ∈ Z ⁿ \ {0} is called an integer relation for the real numbers α ₁,...,α _n , if Σα _i m _i =0 holds. We present an algorithm that when given algebraic numbers α ₁,...,α _n and a parameter ɛ either finds an integer relation for α ₁,...,α _n or proves that no relation of euclidean length shorter than 1/ɛ exists. Each algebraic number is assumed to be given by its minimal polynomial and by a rational approximation precise enough to separate it from its conjugates.

Our algorithm uses the Lenstra-Lenstra-Lovász lattice basis reduction technique. It performs

$$poly \left( {log 1/\varepsilon ,n, log max_i height\left( {\alpha _i } \right), \left[ {Q\left( {\alpha _1 ,...,\alpha _n } \right):} \right]Q} \right)$$

bit operations. The straightforward algorithm that works with a primitive element of the field extension Q(α ₁,...,α _n ) of Q would take poly (n, log max_i height(α _i ), Π ⁱ⁼¹_n degree (α _i )) bit operations.