  We complete the complexity classification by degree of minimizing a
polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous
work shows that optimizing a quadratic polynomial over the integer points in a
polyhedral region in $\mathbb{R}^2$ can be done in polynomial time, while
optimizing a quartic polynomial in the same type of region is NP-hard. We close
the gap by showing that this problem can be solved in polynomial time for cubic
polynomials.
  Furthermore, we show that the problem of minimizing a homogeneous polynomial
of any fixed degree over the integer points in a bounded polyhedron in
$\mathbb{R}^2$ is solvable in polynomial time. We show that this holds for
polynomials that can be translated into homogeneous polynomials, even when the
translation vector is unknown. We demonstrate that such problems in the
unbounded case can have smallest optimal solutions of exponential size in the
size of the input, thus requiring a compact representation of solutions for a
general polynomial time algorithm for the unbounded case.
