Re-Analysis of Polarization in the Gamma-ray flux of GRB 021206

(abridged) A previous analysis of the Reuven Ramaty High Energy Solar Spectroscopic Imager observation of GRB 021206 found that the gamma-ray flux was 80+/-20 polarized. We re-examine this data and find no signal that can be interpreted as due to polarization. First, we find that the number of scattering events suitable for measuring polarization -- having been scattered from one detector to another, with a count produced in both -- is considerably lower than estimated by Coburn and Boggs (2003; CB03), by a factor of 10. The signal-to-noise of the data-set is thus too low to produce a detection, even from a 100% polarized source. Nonetheless, we develop a polarization-detection analysis limited in sensitivity only by Poisson noise, which does not require a space-craft mass model to detect polarization, as in CB03. We find no signal which might be interpreted as due to polarization. Separately, we reproduce the CB03 signal and show that it is not due to polarization. Rather, the CB03 signal is consistent with the previously-neglected systematic uncertainty in the ``null lightcurve'' used for detection. Due to the low signal-to-noise ratio of the RHESSI data, our Poisson noise-limited analysis results in an upper limit consistent with 100%-polarization of the gamma-ray flux from GRB 021206. Thus, no observational constraint on the polarization of GRB 021206 can be derived from these data.