A short proof of the existence of vector Euclidean algorithms
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- by Helaman Ferguson PDF
- Proc. Amer. Math. Soc. 97 (1986), 8-10 Request permission
Abstract:
The classical Euclidean algorithm for pairs of real numbers is generalized to real $n$-vectors by ${\text {Alg}}\left ( {n,{\mathbf {Z}}} \right )$. An iteration of ${\text {Alg}}\left ( {n,{\mathbf {Z}}} \right )$ is defined by three steps. Given $n$ real numbers ${\text {Alg}}\left ( {n,{\mathbf {Z}}} \right )$ constructs either $n$ coefficients of a nontrivial integral linear combination which is zero or $n$ independent sets of simultaneous approximations. Either the coefficients will be a column of a ${\text {GL}}\left ( {n,{\mathbf {Z}}} \right )$ matrix or the simultaneous approximations will be rows of ${\text {GL}}\left ( {n,{\mathbf {Z}}} \right )$ matrices constructed by ${\text {Alg}}\left ( {n,{\mathbf {Z}}} \right )$. This algorithm characterizes linear independence of reals over rationals by ${\text {GL}}\left ( {n,{\mathbf {Z}}} \right )$ orbits of rank $n - 1$ matrices.References
- H. R. P. Ferguson and R. W. Forcade, Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no.Β 6, 912β914. MR 546316, DOI 10.1090/S0273-0979-1979-14691-3
- H. R. P. Ferguson and R. W. Forcade, Multidimensional Euclidean algorithms, J. Reine Angew. Math. 334 (1982), 171β181. MR 667456
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 97 (1986), 8-10
- MSC: Primary 11H46; Secondary 11J13, 11Y16
- DOI: https://doi.org/10.1090/S0002-9939-1986-0831375-X
- MathSciNet review: 831375