  We consider the efficacy of using luminosity distance measurements of deep
redshift supernovae to discriminate between two forms of dark energy,
quintessence (a scalar field with canonical kinetic terms rolling down a
potential) and k-essence (a scalar field whose cosmic evolution is driven
entirely by non-linear kinetic terms). The primary phenomenological distinction
between the two types of models that can be quantified by supernova searches
(at least in principle) is that the equation of state $w\equiv p/\rho$ of
quintessence is falling today while that of k-essence is rising. By simulating
$10^5$ possible datasets that SNAP could obtain, we show that even if the mass
density $\Omega_m$ is known exactly, an ambiguity remains that may not allow a
definitive distinction to be made between the two types of theories.
