  Many combinatorial properties of a point set in the plane are determined by
the set of possible partitions of the point set by a line. Their essential
combinatorial properties are well captured by the axioms of oriented matroids.
In fact, Goodman and Pollack (Journal of Combinatorial Theory, Series A, Volume
37, pp. 257-293, 1984) proved that the axioms of oriented matroids of rank $3$
completely characterize the sets of possible partitions arising from a natural
topological generalization of configurations of points and lines. In this
paper, we introduce a new class of oriented matroids, called degree-$k$
oriented matroids, which captures essential combinatorial properties of the
possible partitions of point sets in the plane by the graphs of polynomial
functions of degree $k$. We prove that the axiom of degree-$k$ oriented
matroids completely characterizes the sets of possible partitions arising from
a natural topological generalization of configurations formed by points and the
graphs of polynomial functions degree $k$. It turns out that the axiom of
degree-$k$ oriented matroids coincides with the axiom of ($k+2$)-signotopes,
which was introduced by Felsner and Weil (Discrete Applied Mathematics, Volume
109, pp. 67-94, 2001) in a completely different context. Our result gives a
two-dimensional geometric interpretation for ($k+2$)-signotopes and also for
single element extensions of cyclic hyperplane arrangements in
$\mathbb{R}^{n-k-3}$.
