Computing the dimension of a semi-algebraic set

In this paper, we consider the problem of computing the real dimension of a given semi-algebraic subset of $\mathbf{R}^k$, where $\mathbf{R}$ is a real closed field. We prove that the dimension, $k'$, of a semi-algebraic set described by $s$ polynomials of degree $d$ in $k$ variables can be computed in time
$$ \begin{cases} s^{(k-k')k'}d^{O(k'(k-k'))},&\text{if}\ k'\geqslant k/2,\\ s^{(k-k'+1)(k'+1)}d^{O(k'(k-k'))},&\text{if}\ k'< k/2. \end{cases} $$
This result improves slightly the result proved in [22], where an algorithm with complexity bound $(sd)^{O(k'(k-k'))}$ is described for the same problem. The complexity bound of the algorithm described in this paper has a better dependence on the number, $s$, of polynomials in the input.