Incrementally and inductively constructing basis of multiplicative dependence lattice of non-zero algebraic numbers
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Let x=(x_1,x_2,...,x_n)^T be a vector of non-zero algebraic numbers, the set R_x:={(k_1,k_2,...,k_n)^T∈Z^n | x_1^k_1x_2^k_2... x_n^k_n=1} is called the multiplicative dependence lattice of x. This paper develops an efficient incremental algorithm to compute a basis of R_x. This algorithm constructs inductively a basis of the lattice as the dimension increases. This is the very first algorithm for computing the basis of the lattice, although a lot of efforts have been made to understand this lattice. In this paper we propose the conception of the rank of a finite sequence of non-zero algebraic numbers, which turns out to be closely related to the rank of the lattice, and as well as to the complexity. The complexity of the algorithm depends not mainly on the dimension n but on the rank of the sequence x_1,x_2,...,x_n, which can be much smaller than n.
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