IJPAM: Volume 47, No. 2 (2008)

APPLICATIONS OF TRANSLATIONS ON $\mathbf{C}^{3}$ AND
$\mathbf{C}^{4}$ TO MOTION IN REAL SPACE

Kristofer Jorgenson
Department of Mathematics and Computer Science
Sul Ross State University
Box C-18, Alpine, Texas, 79832, USA
e-mail: kjorgenson@sulross.edu


Abstract.Translations of $\mathbf{C}^{3}$ and $\mathbf{C}^{4}$ can be used to model the motion of points in $\mathbf{R}^{2}$ and $\mathbf{R}^{3}$, respectively. For $n=2,3$, given a set of paired points in $\mathbf{R}%
^{n}$ that satisfy minimal conditions, it is proved that a fixed-point free, additive group action on $\mathbf{C}^{n+1}$ exists that contains the given points and restricts to a real action on real space. It is known that the action on $\mathbf{C}^{3}$ is equivariantly trivial, and if further conditions are satisified, the action on $\mathbf{C}^{4}$ is similarly equivalent to a translation. It will be shown in each case that these actions will induce translations on $\mathbf{R}^{2}$ and $%
\mathbf{R}^{3}$, respectively. Detailed examples are provided and additional questions are asked.

Received: June 11, 2008

AMS Subject Classification: 14L30, 13A50

Key Words and Phrases: group actions on varieties, actions of groups on commutative rings, invariant theory

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 47
Issue: 2