In this paper, we focus on computing local minimizers of a multivariate polynomial optimization problem under certain genericity conditions. Using a technique from computer algebra and the second-order optimality condition, we provide a univariate representation for the set of local minimizers. In particular, for the unconstrained problem, i.e., the constraint set is \({{\,\mathrm{\mathbb {R}}\,}}^n\) , the coordinates of all local minimizers can be represented by the values of n univariate polynomials at the real solutions of a univariate system containing a polynomial equation and a polynomial matrix inequality. We also develop the technique for problems with equality/inequality constraints. Based on the above technique, we design algorithms to enumerate the local minimizers and provide some experimental examples based on hybrid symbolic-numerical computations. For the case that the genericity conditions fail, at the end of the paper we propose a perturbation technique to compute approximately a global minimizer, provided that the constraint set is compact.
