  In this paper, we investigate the complexity of the central path of
semidefinite optimization through the lens of real algebraic geometry. To that
end, we propose an algorithm to compute real univariate representations
describing the central path and its limit point, where the limit point is
described by taking the limit of central solutions, as bounded points in the
field of algebraic Puiseux series. As a result, we derive an upper bound
$2^{O(m+n^2)}$ on the degree of the Zariski closure of the central path, when
$\mu$ is sufficiently small, and for the complexity of describing the limit
point, where $m$ and $n$ denote the number of affine constraints and size of
the symmetric matrix, respectively. Furthermore, by the application of the
quantifier elimination to the real univariate representations, we provide a
lower bound $1/\gamma$, with $\gamma =2^{O(m+n^2)}$, on the convergence rate of
the central path.
