  In a previous work we proved that each $n$-dimensional convex polyhedron
${\mathcal K}subset{\mathbb R}^n$ and its relative interior are regular images
of ${\mathbb R}^n$. As the image of a non-constant polynomial map is an
unbounded semialgebraic set, it is not possible to substitute regular maps by
polynomial maps in the previous statement. In this work we determine
constructively all unbounded $n$-dimensional convex polyhedra ${\mathcal
K}\subset{\mathbb R}^n$ that are polynomial images of ${\mathbb R}^n$. We also
analyze for which of them the interior ${\rm Int}({\mathcal K})$ is a
polynomial image of ${\mathbb R}^n$. A discriminating object is the recession
cone $\vec{\mathcal C}({\mathcal K})$ of ${\mathcal K}$. Namely, \em ${\mathcal
K}$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal
C}}({\mathcal K})$ has dimension $n$\em. In addition, \em ${\rm Int}({\mathcal
K})$ is a polynomial image of ${\mathbb R}^n$ if and only if $\vec{{\mathcal
C}}({\mathcal K})$ has dimension $n$ and ${\mathcal K}$ has no bounded faces of
dimension $n-1$\em. A key result is an improvement of Pecker's elimination of
inequalities to represent semialgebraic sets as projections of algebraic sets.
Empirical approaches suggest us that there are `few' polynomial maps that have
a concrete convex polyhedron as a polynomial image and that there are even
fewer for which it is affordable to show that their images actually correspond
to our given convex polyhedron. This search of a `needle in the haystack'
justifies somehow the technicalities involved in our constructive proofs.
