Is the dark energy equation of state parameter singular?
  A dark energy with a negative energy density in the past can simultaneously
address various cosmological tensions, and if it is to be positive today to
drive the observed acceleration of the universe, we show that, it should have a
pole in its equation of state parameter. More precisely, in a spatially uniform
universe, a perfect fluid (submitting to the usual continuity equation of local
energy conservation) whose energy density $\rho(z)$ vanishes at an isolated
zero $z=z_p$, necessarily has a pole in its equation of state parameter $w(z)$
at $z_p$, and, $w(z)$ diverges to positive infinity in the limit $z\to z_p^+$
and it diverges to negative infinity in the limit $z\to z_p^-$ -- we assume
that $z_p$ is not an accumulation point for poles of $w(z)$. However, the
converse statement that this kind of a pole of $w(z)$ corresponds to a
vanishing energy density at that point, is not true as we show by a
counterexample.
