Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two
Helaman Rolfe Pratt Ferguson, Rodney W. Forcade · Bulletin of the American Mathematical Society · 1979
A construction using integral matrices with determinant ± 1 is given which has as corollaries generalizations of classical theorems of Dirichlet and Kronecker.This construction yields a geometrically convergent algorithm successfully generalizing the Euclidean algorithm to finite sets of real numbers.Applied to such a set this algorithm terminates if and only if the set is integrally linearly dependent and the algorithm gives absolute simultaneous integral approximations if and only if the set is integrally linearly independent.This development applies to complex numbers, can be used to give proofs of irreducibility of polynomials and yields effective lower bounds on heights of integral relations.