In this paper, we study the computation of the $\mathcal{L}_{\infty}$-norm for finite-dimensional linear systems. This problem is first reduced to the computation of the maximal $x$-projection of the real solutions $(x, y)$ of a bivariate polynomial system $\{\mathcal{P},\frac{\partial \mathcal{P}}{\partial y}\} \subset \mathbb{Z}[x,y]$. We then apply computer algebra methods to solve the problem. We alternatively study a method based on rational univariate representations, a method based on root separation, and finally a method based on the sign variation of the leading coefficients of the signed subresultant sequence and on the identification of an isolating interval for the maximal $x$-projection of the real solutions of the system.