# Integer relations among algebraic numbers

• Bettina Just
Communications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 379)

## Abstract

A vector m=(m1,...,m n ) ∈ Zn \ {0} is called an integer relation for the real numbers α1,...,α n , if Σα i m i =0 holds. We present an algorithm that when given algebraic numbers α1,...,α n and a parameter ɛ either finds an integer relation for α1,...,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(α1,...,α n ) of Q would take poly (n, log maxi height(α i ), Π n i=1 degree (α i )) bit operations.

## Keywords

Integer relation algebraic number lattice basis reduction

## MSC

68Q25 68Q40 12F10

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