On combinatorial properties of points and polynomial curves

Oriented matroids are a combinatorial model, which can be viewed as a combinatorial abstraction of partitions of point sets in the Euclidean space by families of hyperplanes. They capture essential combinatorial properties of geometric objects such as point configurations, hyperplane arrangements, and polytopes. In this paper, we introduce a new class of oriented matroids, called degree-$k$ oriented matroids, which captures the essential combinatorial properties of partitions of point sets in the $2$-dimensional Euclidean space by graphs of polynomial functions of degree $k$. We prove that the axioms of degree-$k$ oriented matroids completely characterize combinatorial structures arising from a natural geometric generalization of configurations formed by points and graphs of polynomial functions degree $k$. This may be viewed as an analogue of the Folkman-Lawrence topological representation theorem for oriented matroids.Comment: 15 page