  In this paper we consider the classical problem of computing linear
extensions of a given poset which is well known to be a difficult problem.
However, in our setting the elements of the poset are multivariate polynomials,
and only a small "admissible" subset of these linear extensions, determined
implicitly by the evaluation map, are of interest. This seemingly novel problem
arises in the study of global dynamics of gene regulatory networks in which
case the poset is a Boolean lattice. We provide an algorithm for solving this
problem using linear programming for arbitrary partial orders of linear
polynomials.
  This algorithm exploits this additional algebraic structure inherited from
the polynomials to efficiently compute the admissible linear extensions. The
biologically relevant problem involves multilinear polynomials and we provide a
construction for embedding it into an instance of the linear problem.
