High-redshift cosmography: auxiliary variables versus Pad\'e polynomials
  Cosmography becomes non-predictive when cosmic data span beyond the red shift
limit $z\simeq1 $. This leads to a \emph{strong convergence issue} that
jeopardizes its viability. In this work, we critically compare the two main
solutions of the convergence problem, i.e. the $y$-parametrizations of the
redshift and the alternatives to Taylor expansions based on Pad\'e series. In
particular, among several possibilities, we consider two widely adopted
parametrizations, namely $y_1=1-a$ and $y_2=\arctan(a^{-1}-1)$, being $a$ the
scale factor of the Universe. We find that the $y_2$-parametrization performs
relatively better than the $y_1$-parametrization over the whole redshift
domain. Even though $y_2$ overcomes the issues of $y_1$, we get that the most
viable approximations of the luminosity distance $d_L(z)$ are given in terms of
Pad\'e approximations. In order to check this result by means of cosmic data,
we analyze the Pad\'e approximations up to the fifth order, and compare these
series with the corresponding $y$-variables of the same orders. We investigate
two distinct domains involving Monte Carlo analysis on the Pantheon Superovae
Ia data, $H(z)$ and shift parameter measurements. We conclude that the (2,1)
Pad\'e approximation is statistically the optimal approach to explain low and
high-redshift data, together with the fifth-order $y_2$-parametrization. At
high redshifts, the (3,2) Pad\'e approximation cannot be fully excluded, while
the (2,2) Pad\'e one is essentially ruled out.
