  Multiobjective discrete programming is a well-known family of optimization
problems with a large spectrum of applications. The linear case has been
tackled by many authors during the last years. However, the polynomial case has
not been deeply studied due to its theoretical and computational difficulties.
This paper presents an algebraic approach for solving these problems. We
propose a methodology based on transforming the polynomial optimization problem
in the problem of solving one or more systems of polynomial equations and we
use certain Gr\"obner bases to solve these systems. Different transformations
give different methodologies that are analyzed and compared from a theoretical
point of view and by some computational experiments via the algorithms that
they induce.
