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 point configurations, hyperplane arrangements, and polytopes, and oriented matroid theory has been developed in the context of various research fields.
In this paper, we introduce a new class of oriented matroids, called degree- oriented matroids, which captures the essential combinatorial properties of partitions of point sets in the -dimensional Euclidean space by graphs of polynomial functions of degree . We prove that the notion of degree- oriented matroids completely characterizes combinatorial structures arising from a natural geometric generalization of configurations formed by points and graphs of polynomial functions degree . This may be viewed as an analogue of the Folkman-Lawrence topological representation theorem for oriented matroids.
Oriented matroids are a combinatorial model, introduced independently by Bland, Folkman, Las Vergnas, and Lawrence (published in  and ). Oriented matroids abstract various objects such as point configurations, vector configurations, polytopes, hyperplane arrangements, and digraphs, and provide a unified framework for discussing various combinatorial aspects of these objects. They have a rich structure, which can describe, for example, face structures of polytopes and hyperplane arrangements, partitions of point sets by hyperplanes, linear programming duality, and Gale duality in polytope theory. Nowadays, oriented matroid theory is a fairly rich subject of research, with connections to various research fields such as discrete and computational geometry, graph theory, operations research, topology, and algebraic geometry (see ). One of the outstanding results in oriented matroid theory is the topological representation theorem, introduced by Folkman and Lawrence  (see [2, 4, 5] for simplified proofs), which says that there is a one-to-one correspondence between oriented matroids and equivalence classes of pseudosphere arrangements. This theorem bridges topology and combinatorics, and shows that oriented matroids constitute a natural combinatorial model.
Motivated by oriented matroid theory, it is natural to consider whether a useful theory can be developed by abstracting combinatorial aspects of other related objects. Because one of the common ways of understanding oriented matroids is the combinatorial abstraction of partitions of point sets in the Euclidean space by families of hyperplanes, we consider combinatorial abstraction of partitions of point sets by other geometric objects.
Recently, Eliáš and Matoušek  proposed a new interesting generalization of the Erdős-Szekeres theorem. To prove the theorem, they investigated the combinatorial properties of partitions of point sets by graphs of certain families of polynomial functions of degree . In this paper, we discuss more closely what kind of combinatorial properties are exhibited by such partitions. In fact, we show that the combinatorial properties observed by Eliáš and Matoušek represent in some sense almost all of the combinatorial properties that can be proved using some natural geometric properties. To do so, we first observe a slightly stronger combinatorial property of partitions than those observed by Eliáš and Matoušek, and then we introduce a combinatorial model called degree- oriented matroids by axiomatizing the observed combinatorial properties. We prove that degree- oriented matroids completely characterize partitions arising from -intersecting pseudoconfigurations of points, which are a natural geometric generalization of configurations of points and graphs of polynomial functions of degree . This provides an analogue of the topological representation theorem for oriented matroids, and shows that degree- oriented matroids capture the essential combinatorial properties of partitions of point sets by graphs of polynomial functions of degree .
The partitions of a point set in the Euclidean space by a certain family of spheres determine an oriented matroid, which is called a Delaunay oriented matroid (see [2, Section 1.9]). Santos  proved that partitions of a point set in the 2-dimensional Euclidean space by spheres defined by a smooth, strictly convex distance function also fulfill the oriented matroid axioms. Miyata  proved that the class of partitions of point sets in the 2-dimensional Euclidean space by pseudocircles coincides with the rank matroid polytopes. Ardila and Develin  introduced tropical oriented matroids as a combinatorial abstraction of tropical hyperplane arrangements. Horn  proved that every tropical oriented matroid can be represented as a tropical pseudohyperplane arrangement.
Here, we summarize the notation that will be employed in this paper. In the following, we assume that is a finite ordered set, is a sign vector on , is a point in the -dimensional Euclidean space, and is a positive integer.
: the -coordinate of .
: the -coordinate of .
In this section, we summarize some basic facts regarding oriented matroids. See  for further details.
Let be a point configuration in general position (i.e., no points of lie on the same hyperplane) in , and let be the vector configuration in with . To see how is separated by hyperplanes, let us consider the map defined by
where (resp. ) if (resp. ). Then, we have
The map contains rich combinatorial information regarding , such as convexity, the face lattice of the convex hull, and possible combinatorial types of triangulations (see ). The chirotope axioms of oriented matroids are obtained by abstracting the properties of .
(Chirotope axioms for oriented matroids)
For and a finite set , a map is called a chirotope if it satisfies the following axioms. The pair is called an oriented matroid of rank .
is not identically zero.
, for any and any permutation on .
For any , we have
We remark that (B3) is combinatorial abstraction of the Grassmann-Plücker relations:
where for .
The set also contains equivalent information to . The cocircuit axioms of oriented matroids are introduced by abstracting the properties of the set .
(Cocircuit axioms for oriented matroids)
A collection satisfying Axioms (C0)–(C3) is called the set of cocircuits of an oriented matroid.
For all , if , then or .
For any and , there exists with
, , and .
From a chirotope , we can construct the cocircuits . It is also possible to reconstruct the chirotope (up to a sign reversal) from the cocircuits . A rank oriented matroid is said to be uniform if for any (equivalently if for any cocircuit ). If the underlying strucure of the set is known to be uniform, i.e., if for any , then Axiom (C3) can be replaced by the following axiom:
(C3’) For any with and , there exists with , , and .
More generally, Axiom (C3) can be replaced by the axiom of modular elimination. For further details, see [2, Section 3.6].
An oriented matroid is acyclic if for any there exists with and .
It can easily be seen that oriented matroids arising from point configurations are acyclic.
One of the outstanding facts in oriented matriod theory is that oriented matroids always admit topological representations, as established by Folkman and Lawrence . Here, we explain a variant of this fact, which was originally formulated in terms of allowable sequences by Goodman and Pollack . First, we observe that oriented matroids also arise from generalization of point configurations, called pseudoconfigurations of points (also called generalized configurations of points).
(Pseudoconfigurations of points)
A pair of a point configuration in and a collection of unbounded Jordan curves is called a pseudoconfiguration of points (or a generalized configuration of points) if the following hold.
For any , there exist at least two points of lying on .
For any two points of , there exists a unique curve in that contains both points.
Any pair of (distinct) curves intersects at most once.
For each , we label the two connected components of arbitrarily as and . Then, we assign the sign vector such that , and , and we let . Then, turns out to be an acyclic oriented matroid of rank . Goodman and Pollack  proved that in fact the converse also holds.
(Topological representation theorem for acyclic oriented matroids of rank )
For any acyclic oriented matroid of rank , there exists a pseudoconfiguration of points with .
Here, the assumption that is acyclic is not important, because non-acyclic oriented matroids can be represented as signed pseudoconfigurations of points, where each point has a sign.
The notion of pseudoconfigurations of points in the -dimensional Euclidean space can be introduced analogously, but not every acyclic oriented matroid of rank can be represented as a pseudoconfiguration of points in the -dimensional Euclidean space. For further details, see [2, Section 5.3]. However, oriented matroids of general rank can be represented as pseudosphere arrangements , with further details presented in [2, Section 5.2].
3 Definition of degree- oriented matroids
In this section, we introduce degree- oriented matroids as a combinatorial model, which captures the essential combinatorial properties of configurations of points and graphs of polynomial functions. To do so, we first review some results that were introduced by Eliáš and Matoušek .
Eliáš and Matoušek  introduced the notion of th-order monotonity, which is a generalization of the usual notion of monotonity, described as follows. (Because it is more convenient in our context to reinterpret the st-order monotonity of Eliáš and Matoušek as the th-order mononity, the following definitions are slightly different from the originals.) Let be a point configuration in the -dimensional Euclidean space with . We assume that is in -general position, i.e., no points of lie on the graph of a polynomial function of degree at most . Furthermore, we define a -tuple of points in to be positive (resp. negative) if they lie on the graph of a function whose ()st order derivative is everywhere nonnegative (resp. nonpositive). A subset is said to be th-order monotone if its -tuples are either all positive or all negative. This notion can be stated in an alternative manner using the map , defined as follows.
where is the Newton interpolation polynomial of the points , i.e.,
is the unique polynomial function of degree whose graph passes through the points .
Under this definition, a subset is th-order monotone if and only if
for all (see [6, Lemma 2.4]).
This map contains information on which side of the graph of the polynomial function of degree determined by the point lies.
In other words, the map contains information regarding the partitions of by graphs of polynomial functions of degree .
It is shown in  that the map can be computed in terms of determinants of a higher-dimensional space.
(-dimensional linear representability [6, Lemma 5.1])
Let be the map that sends each point to . Then, we have
i.e., the map is a chirotope of an oriented matroid of rank .
Eliáš and Matoušek  additionally observed the following useful property.
(Transitivity [6, Lemma 2.5])
If for , then we have for all .
This proposition is proved in , using Newton interpolation polynomials. Here, we provide a geometric proof for . The generalization of this is straightforward.
Without loss of generality, we assume that , which means that the point is above the graph . Note that the graphs and intersect at and , and that they do not intersect elsewhere. This indicates that the graph lies above the graph for . The point is above the graph by the assumption , and it follows that lies above the graph , which implies that . The same argument can be applied when either of the graphs or are considered instead of .
In , this property is used to prove a new generalization of the Erdős-Szekeres theorem, which states that there is always a -monotone subset of size in any point set of size in -general position, where is the th tower function. Our first observation is that actually admits the following stronger property.
(()-locally unimodal property)
For any and with (lexicographic order, i.e., there exist () with ), it holds that if , then .
In the next section, we will prove that the above-mentioned properties are in some sense all combinatorial properties that can be proved using some natural geometric properties. This motivates us to consider combinatorial structures characterized by those properties.
(Degree- oriented matroids)
Let be a finite ordered set. We say that a rank uniform oriented matroid is called a degree- uniform oriented matroid if it satisfies the following condition. For any and with (lexicographic order), it holds that if , then .
4 Geometric representation theorem for degree- oriented matroids
In this section, we prove that degree- oriented matroids can always be represented by the following generalization of configurations formed by points and graphs of polynomial functions of degree .
(-intersecting pseudoconfiguration of points)
A pair of a point configuration () in the 2-dimensional Euclidean space and a collection of -monotone Jordan curves is called a -intersecting pseudoconfiguration of points if the following conditions hold:
For any , there exist at least points of lying on .
For any points of , there exists a unique curve passing though each point.
For any (), and intersect (transversally) at most times.
Here, a Jordan curve is called -monotone if it intersects with any vertical line at most once.
For , we denote .
If for all , then the configuration is said to be simple.
When is a simple configuration, we denote the curve determined by points by .
An -monotone Jordan curve can be written as , for some continuous function .
We define , , and , .
We call (resp. ) the -side (resp. -side) of .
For , the subconfiguration induced by is a -intersecting pseudoconfiguration ,
For each -intersecting pseudoconfigurations of points, a partition function is defined in a similar manner as in the case of configurations formed by points and graphs of polynomial functions of degree . To define and analyze this, we require the following notion.
Let be a -intersecting pseudoconfiguration of points. A lens (or full lens) of is a region that can be represented as , where , and and are the -coordinates of consecutive intersection points of and . Note that consists of two points. We call these the end points of . The end point with the greater (resp. smaller) -coordinate is called the right (resp. left) end point, and is denoted by (resp. ).
A half lens of is a region represented as or , where , and and are the -coordinates of the leftmost and rightmost intersection points of and , respectively. We call a full or half lens an empty lens if .
Given a simple -intersecting pseudoconfiguration of points (or more generally a configuration satisfying only (PP1) and (PP2)), we define a map as follows.
For with , we have
if for some ().
for any and any permutation on .
The map is well-defined.
It suffices to show that for any permutation on with and any sign , we have . First, we pick such a arbitrarily. Then, we have
for some and . If , then the proposition is trivial, and thus we assume that . Then, the curves and intersect at the points . By the condition of -intersecting pseudoconfigurations of points, these two curves must not intersect elsewhere, and thus the curve must lie above over the interval if is above . Because the above-below relationship of and is reversed at each end point of each lens formed by these two curves, the curve must lie above (resp. below) over the interval if is even, i.e., if (resp. if is odd, i.e., if ). Therefore, we have . The same discussion applies in the case that is below .
Step 1: -intersecting pseudoconfigurations of points determine degree- oriented matroids
For every simple -intersecting pseudoconfiguration of points , the map defines a degree- uniform oriented matroid.
First, we prove that the map is a chirotope of an oriented matroid of rank . For this, it suffices to prove that the set fulfills the cocircuit axioms of oriented matroids. Clearly, Axioms (C0)–(C2) are satisfied and we need only verify Axiom (C3). Because we have for all , it suffices to verify Axiom (C3’). Take with . Let be sign vectors that correspond to and , respectively ( and are determined uniquely up to a sign reversal). Take an and any , and let be the sign vector with that corresponds to . Let us verify that is a required cocircuit in (C3’). Note that and form lenses (see Figure 5). Because already intersects with and times at the points with indices in , it cannot intersect with or elsewhere. Therefore, if is contained inside of one of the lenses, then the whole of must lie in the lenses. Take any with . Then, lies outside of the lenses formed by and , and thus and lie on the same side of . If is outside of the lenses, then the whole of must lie outside of the lenses (except for the end points). Points with corresponds to points inside of the lenses and a similar discussion shows that . Therefore, we have and .
Next, we confirm the ()-locally unimodal property. Suppose that there exist and such that and , . Let and () be the integers such that , , and . Let be the integer such that , where we assume that and . Let the integers and be defined similarly. Since , the point is on the -side of and on the -side of . Because and must not intersect more than times, the curves and form lenses with end points , and must lie inside of these lenses. Now, we remark that the point is on the -side of and the -side of , because and . Therefore, the curve must intersect with either or in . This means that must intersect with either or at least times, which is a contradiction.
Step 2: Degree- oriented matroids can be represented as -intersecting pseudoconfigurations of points
Here, we prove that every degree- oriented matroid admits a geometric representation as a -intersecting pseudoconfiguration of points. To this end, we introduce two operations.
(Empty lens elimination I)
Let be an empty lens represented as , where , and (resp. ) can be (resp. ). Transform and by connecting , , and (when ), and by connecting , , and (when ), for sufficiently small , so that the new curves do not touch around the vertical lines and (see Figure 7).
(Empty lens elimination II)
Let be lenses represented as and , …, where , and , . Suppose that , , …, and are empty for some . Transform and by connecting , , and , and by connecting , , and , for sufficiently small , so that the new curves do not touch around the vertical lines and (see Figure 8).
The two operations decribed above decrease the total number of intersection points of the curves in , without altering the above-below relationships between the points in and the curves in . However, they may invalidate the condition that each pair of curves in intersect at most times. However, we prove that a -intersecting pseudoconfiguration of points is always obtained after the operations are applied as far as possible.
Let be a simple configuration with a finite point set in and a collection of -monotone Jordan curves satisfying (PP1) and (PP2) in Definition 4.1 (with some pair of curves possibly intersecting more than times). We assume that the map fulfills the axioms of degree- oriented matroids. If it is impossible to apply the empty lens eliminations to , then is a -intersecting pseudoconfiguration of points.
Assume that there is a pair of curves in that intersect more than times. Let be a minimal subset of such that there are two curves in intersecting more than times after the empty lens eliminations are applied as far as possible. Let and be curves in that have the smallest -coordinate for the st intersection point. The curves and form full lenses and two half lenses. Let us label these full and half lenses by in increasing order for the -coordinates. Because the empty lens eliminations I and II cannot be applied, there must exist distinct points with . Note that there are no points outside of the lenses , by the minimality of . Let