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Additive and multiplicative relations connecting conjugate algebraic numbers. (English) Zbl 0586.12001

The author investigates the possibility of solving equations of the form (A) \(\alpha_1^{a_1}\cdots\alpha_k^{a_k}=1\) and (B) \(a_1\alpha_1+\ldots+a_k\alpha_k=0\), where \(a_1,\ldots,a_k\) are given non-zero integers with greatest common factor 1 and where each \(\alpha_ i\) is to be a conjugate of some algebraic integer \(\alpha\) (with the \(\alpha_i\) not necessarily distinct). A solution of (A) (resp. (B)) is trivial if \(\alpha\) is a root of unity (resp. \(\alpha =0)\). He shows that (A) and (B) have a non-trivial solution if and only if there is an \(n\geq k\) and an \(n\times n\) incomplete Latin square of determinant zero, each row and column being a permutation of \(a_1,\ldots, a_k\) and \((n-k)\) zeros. This result has the unexpected consequence that (A) is solvable if and only if (B) is solvable.
He gives a computationally more convenient formulation of the criterion in terms of sets of rational integer solutions to \(a_1x_1+\ldots+a_kx_k=0\) and applies this result to a number of interesting examples.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11D57 Multiplicative and norm form equations
05B15 Orthogonal arrays, Latin squares, Room squares
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