Generalized Dissections and Monsky's Theorem
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GENERALIZED DISSECTIONS AND MONSKY’S THEOREM
AARON ABRAMS AND JAMIE POMMERSHEIM
Abstract.
Monsky’s celebrated equidissection theorem follows from his moregeneral proof of the existence of a polynomial relation f among the areas of thetriangles in a dissection of the unit square. More recently, the authors studieda different polynomial p , also a relation among the areas of the triangles in sucha dissection, that is invariant under certain deformations of the dissection. Inthis paper we study the relationship between these two polynomials.We first generalize the notion of dissection, allowing triangles whose orien-tation differs from that of the plane. We define a deformation space of thesegeneralized dissections and we show that this space is an irreducible algebraicvariety. We then extend the theorem of Monsky to the context of generalizeddissections, showing that Monsky’s polynomial f can be chosen to be invariantunder deformation. Although f is not uniquely defined, the interplay between p and f then allows us to identify a canonical pair of choices for the polyno-mial f . In many cases, all of the coefficients of the canonical f polynomialsare positive. We also use the deformation-invariance of f to prove that thepolynomial p is congruent modulo 2 to a power of the sum of its variables. Introduction
In 1970 Paul Monsky proved the following theorem:
Theorem (Monsky [9]) . Fix a dissection of the unit square into n triangles, anddenote the areas of the triangles by a , . . . , a n . Then there is an integer polynomial f in n indeterminates such that f ( a , . . . , a n ) = 1 / . A corollary is Monsky’s famous “equidissection” theorem: if a square is dissectedinto n triangles of equal area, then n must be even. This follows because there isno integer polynomial in n variables with f ( n , . . . , n ) = when n is odd.Happy 50th birthday, Monsky’s Theorem!In the half-century since its publication, the equidissection theorem has inspireda significant amount of mathematics, including numerous other equidissection theo-rems in the plane, higher dimensional analogs, approximation theorems, and more.Relatively little attention has been focused on the polynomial f , however.A dissection of a square is defined as a finite collection of triangles in the planewhose interiors do not intersect and whose union is the square. Monsky’s theoremis a statement about dissections.Over the years it has occurred to several people to first fix the combinatorics of adissection, and then try to understand which collections of areas are realized by thetriangles. We heard of this approach from Joe Buhler, whose student Adam Robinswrote [11] about it, and from Serge Tabachnikov, whose students Joshua Kantor Key words and phrases.
Triangulation, area relation, equidissection. RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
A BCD (a)
Example 1: p = A − B + C − Df = A + C A B CD E Fu v (b)
Example 2:See text for p and f . Figure 1.
Seminal examples: p evaluates to zero and f evaluates to 1 / p , different than f , associated to certaindissections which also has one variable for each triangle and which vanishes, ratherthan taking the value 1 /
2, on the input ( a , . . . , a n ). By construction p dependsonly on the combinatorics of the dissection, so the same p also vanishes at anytuple of areas arising by deforming the dissection; indeed under the hypotheses ofour theorem the zero set of p is exactly the area variety of the triangulation, whichis the (closure of the) collection of realizable areas. For elementary reasons p isirreducible and homogeneous.The polynomials p and f are our primary objects of study. These polynomialsare closely related, although they have different roles in the theory. In [1] we called p a Monsky polynomial , but here we emphasize the distinction and the interplaybetween the two, so we give them different names: p is the area polynomial and f is the Monsky polynomial .Here are two examples which make numerous appearances throughout the paper.
Example 1.
The dissection in Figure 1a has area polynomial p = A − B + C − D (or its negative), and Monsky polynomial f = A + C (or ˜ f = B + D , or f + p =2 A − B + 2 C − D , etc.). One can easily see that regardless of where the centralvertex is placed, the polynomial p evaluates to zero, and as long as the square hasunit area, f, ˜ f , and f + p all evaluate to 1 /
2. For any square, f evaluates to halfthe total area. Example 2.
Less apparently, the dissection in Figure 1b has p = A + C + E − AC + 2 AE + 2 CE − B − D − F − BD − BF + 2 DF and f = A + C + E + 2 AE + 2 CE + 2 DF + ( A + C + E )( B + D + F ) . Again, regardless of the placement of u and v , p evaluates to zero and f evaluatesto half the area of the square.The fact that p is well-defined essentially reflects the correctness of a heuris-tic dimension count, whereas Monsky’s polynomial f provides number-theoretic(specifically, mod 2) information. However, in Monsky’s theorem a dissection istreated as a static object, and invariance of f under deformation is not guaran-teed. Think of a dissection in which some triangles in the middle, say i and j , have up to sign RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 3
Figure 2.
A dissection (left) and a generalized dissection (right)of a square.areas summing to 1 /
2. Then the polynomial f ( x , . . . , x n ) = x i + x j satisfies theconclusion of Monsky’s theorem, but the sum a i + a j could easily change when thedissection is deformed. One of our main goals is to extend Monsky’s theorem toshow that f can be made deformation-invariant, as it is in the examples we havealready seen.It turns out that the act of deforming a dissection is trickier than it may appearat first glance, and it deserves to be taken seriously. One issue is that the verticesmay be constrained to lie on certain line segments, so in general the vertices cannotmove freely and independently of each other. Another issue is that one is forcedto confront the possibility that triangles might degenerate, or turn upside-down.In the first part of this paper we develop a framework for handling these issues,building on our work in [1]. The main idea is to view a dissection as the image of acertain map which itself has a natural deformation space. We are led to a notion ofa generalized dissection , and we will see that there are generalized dissections thatcannot be deformed back into (classical) dissections. See Figure 2 (right), wherethere are three triangles, one of which is upside-down. Examples like this turn outto be crucial to our theory.Our main theorem about these deformation spaces, which we call X , is thatthey are irreducible rational varieties. This is proved in Section 4. Our proof ismore subtle than we anticipated, because we encountered fundamental issues aboutarrangements of points and lines that required some finesse to mitigate. In Section5 we discuss some questions that arose in this process and their relationship withthe well-studied areas of point/line configurations and oriented matroids.A different method for treating the problem of deformations has been proposedand studied in [7] by Labb´e, Rote, and Ziegler, who were interested in approximat-ing equidissections.In the second part of the paper, with the foundations now established, we areable to investigate p and f . Our main technical results (Theorems Monsky+ andMonsky++) extend Monsky’s theorem to the deformation space X , showing that f can indeed be chosen to be invariant under deformation. That is, we give algebraicversions of the theorem, showing that for any (generalized) dissection, not only dothe areas of the triangles satisfy a polynomial relation, but also the formulas forthe areas satisfy a polynomial relation. Thus we may think of f as a “dynamic”object, as we did already with p .Once both p and f are thusly defined, we are finally able to rigorously explorethe relationships between the two. In Sections 9 and 10 we prove our main results RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- about p and f by exploiting features of each polynomial to deduce informationabout the other.Specifically, recall from Example 1 above that a given dissection has many Mon-sky polynomials. The canonicalness of { p, − p } allows us to define a canonical pair { f, ˜ f } with extra known and conjectured properties; for example these have mini-mal degree among deformation-invariant polynomials satisfying Monsky’s theorem.These polynomials also often have non-negative coefficients, an observation we willreturn to in Section 11.In the other direction the number-theoretic content of f transports to p , givingadditional information about its structure. For instance we show that mod 2, thepolynomial p is congruent to a power of the sum of the variables.We close in Section 12 with a question about equidissections.One of the pleasant features of the present setup is that we minimize the amountof combinatorial information needed to parameterize the deformation space of ageneralized dissection. This information is often implicit in a drawing of the dis-section, and this setup simplifies the computation of p and f relative to what wedid in [1]. The job is still inherently computationally expensive, but the cost nowessentially depends only on the number of triangles in the dissection, and not howmuch degeneracy there is.Many mysteries remain about these polynomials. Contents
1. Introduction 1
Part 1. Deforming dissections
52. Generalized dissections 53. Constrained triangulations 94. The space of drawings 114.1. Definitions and theorem 114.2. Combinatorial irreducibility and drawing orders 134.3. Proof of theorem 154.4. Home field advantage 165. Musings about ˙ X X X Part 2. Area relations
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GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 5
11. Positivity 3212. Equidissections 34acknowledgements 35References 35
Part Deforming dissections
In this part of the paper we develop the language of generalized dissections andconstrained triangulations, which we use to define deformation spaces of dissections.2.
Generalized dissections
Classically, a dissection of a square is a finite collection of triangles in the (Eu-clidean) plane whose interiors do not intersect and whose union is the square. Ourfirst goal here is to give a more general definition that allows for deformations. Westart by setting some terminology.We work in the affine plane C . (The reader who prefers to think of everythingtaking place in R is encouraged to do so; we prove in Section 4.4 that this makesno difference to our theory.)If S is a cyclically ordered finite set S = ( s , . . . , s n ) then we define an edge of S to be any of the ordered pairs ( s i , s i +1 ), with indices taken mod n .A polygon , or n -gon , is a cyclically ordered set of n ≥ C ,called vertices. A 3-gon is also called a triangle ; thus a triangle comes with anorientation. A polygon is totally degenerate if its vertices are collinear, degenerate if it has three consecutive vertices (in the cyclic order) that are collinear, and non-degenerate if no three consecutive vertices are collinear.An abstract polygon , or abstract n -gon , is a 2-cell whose boundary circle consistsof n n D (including degenerate and totally degenerateones) is an abstract polygon whose vertices are labeled by the points of D (in thesame cyclic order). Associated to a family of polygons we can construct an abstract2-dimensional complex from a corresponding family of abstract polygons by gluingtogether along edges: the edge ( v, w ) of one polygon is glued to the edge ( w, v ) ofanother.Notice that we have chosen to label the vertices of the abstract polygons andcomplexes by the points themselves. For example, the vertex of the abstract polygoncorresponding to (1 ,
1) is called (1 , Definition 1.
Let D be a polygon in C . A generalized dissection of D consists ofa finite set Triangles and a finite set
Constraints such that: (1)
Each element of
Triangles is a non-degenerate triangle in C . (2) Each element of
Constraints is a totally degenerate polygon in C , each ofwhose vertices is a vertex of at least one triangle in Triangles(3)
Any two distinct constraints share at most one vertex Another reasonable name for this vertex would have been v (1 , , which has the advantage ofemphasizing the abstract nature of this vertex, but the disadvantage of being clearer. RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
Figure 3.
A dissection and its associated 2-complex. The shadedcells are the poofagons.(4)
The associated 2-complex built from abstract polygons corresponding to theunion of
Triangles and
Constraints is an oriented disk with boundary equalto D . Note that D is allowed to be degenerate or totally degenerate.We think of the elements of Triangles as the triangles in the dissection, exceptnow they are oriented. Elements of Constraints are interpreted as collinearity con-straints; item (3) ensures that the constraints are maximal. The abstract polygonscorresponding to elements of Constraints are called poofagons . (These may or maynot be triangles, but they are not elements of Triangles.) Item (4) implies that wecan interpret the data as the image of a PL map from a cellulated disk into theplane, under which the poofagons have degenerated into line segments. (Not everysuch map gives a generalized dissection though, as illustrated below by Figure 4d.)Often, the sets Triangles and Constraints are implicitly defined by a drawing.For classical dissections this is always the case, as we prove in Proposition 2 below.An example of a classical dissection is shown in Figure 3, along with the 2-complexassociated to the corresponding (implicitly defined) generalized dissection. Thegeneralized dissection has four poofagons, one quadrilateral and the rest triangles,shown as shaded cells.One should acquaint oneself with a few more examples before proceeding. Somebasic ones are shown in Figure 4, and we separately highlight an especially impor-tant one in Figure 5. Example 3 (cf. Example 1) . Figure 4a is a dissection with four triangles; it isalso a generalized dissection with the same four triangles (now oriented) and noconstraints. The corresponding abstract triangles glue together to form a simplicialcomplex of which this is a drawing.Figure 4b resembles 4a, except the central vertex has been dragged outside thesquare. Here and elsewhere, we have indicated the vertices with small dots in orderto avoid potential confusion with edges that intersect at points of the plane thatare not vertices. This is not a dissection. It is a generalized dissection with fourtriangles, one of which is oriented differently from the plane. As in Figure 4a, thereare no constraints. The corresponding abstract triangles form the same simplicialcomplex as Figure 4a.
Example 4 (cf. Example 2) . Figure 4c is a dissection, both classical and gener-alized, with four triangles. The generalized dissection has a constraint, which is
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GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 7 (a)
A dissec-tion. (b)
A generalizeddissection that isnot a dissection. (c)
A dissec-tion with aconstraint. (d)
Not a dissection.
Figure 4.
Some basic examplesa (totally degenerate) quadrilateral. The associated cell complex can be triangu-lated in two ways by choosing a diagonal of this quadrilateral; one of the resultingsimplicial complexes is shown later in Figure 7.
Example 5.
Figure 4d is not a generalized dissection at all. Although its faces“cancel,” the flattened tetrahedron pinned to the center of the square makes itimpossible to describe this as a generalized dissection. In particular, the simplicialcomplex made from the obvious eight (abstract) triangles is homeomorphic to theone-point union of a disk and a sphere.
Example 6 (The
ACE example, cf. Example 2) . Finally, Figure 5a is a gener-alized dissection with three triangles and three constraints. It is an interestingspecimen. It takes a moment to identify the triangles and the constraints (with thecorrect orientations). There are three triangles, one of which is upside-down. Thereader should verify that this this does indeed satisfy the definitions of a general-ized dissection, with the associated simplicial complex shown in Figure 5b, withthe poofagons shaded. (This is the same 2-complex shown later, in Figure 7.) Onefeature of this example is that it cannot be deformed into a (classical) dissection inwhich all three triangles remain alive.
Proposition 2 ( D ; D ) . The triangles of any (classical) dissection of a square,when oriented counterclockwise in R , comprise the set Triangles of a generalizeddissection.Proof.
Let D be a dissection, and let Triangles be the set of triangles, each orientedcounterclockwise. We need only to specify the collinearity constraints.Recall that D consists of triangles in R . Say that a vertex of D is constrained if it is in the interior of an edge of (a triangle of) D and unconstrained otherwise.Define a segment of D to be a line segment in the plane that is contained in theunion of the boundaries (edges) of the triangles of D and contains no unconstrained RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT -- (a) (b)
Figure 5.
The
ACE example: a generalized dissection with con-straints. This cannot be deformed to a dissection.vertex in its interior. Finally a segment is called maximal if it is not contained inany larger segment and it contains at least three vertices of D .For each maximal segment M , we define a constraint containing exactly thosevertices that are contained in M . To determine the cyclic order, we use the factthat every vertex v in the interior of M is constrained, so all edges containing such v (and not contained in M ) are on the same side of M . Precisely, we traverse theboundary of a small regular neighborhood of M in the plane, counterclockwise.Each time we cross an edge of D , we record the vertex in M that the edge contains.After eliminating duplicates, we have a cyclic ordering on the vertices contained in M . The set Constraints consists of the cyclically ordered sets constructed in thisway.It is now easy to see that we have a generalized dissection. Item (3) of thedefinition is satisfied since constraints intersect exactly where the correspondingmaximal segments intersect, and two such segments cannot overlap in an intervalby maximality. Item (4) of the definition is also satisfied because the associated2-complex is made by cutting the square open along the maximal segments andgluing in poofagons corresponding to the constraints. Definition 3.
A generalized dissection is generic if no line in C contains twointersecting constraints. Example 7.
Figure 6 shows a non-generic dissection D on the left. The vertex v is unconstrained; our definition of generalized dissection does not allow us tointerpret the entire horizontal segment containing v as a single constraint. Insteadwe view this segment as two separate constraints intersecting at v ; this violatesthe definition of generic. The middle and right figures show two generic dissectionsthat are close to D . The middle figure has two constraints, whereas on the rightthe constraints have been merged and there is an additional vertex. We will see inSection 4 that these two generic dissections have different deformation spaces.In Section 4 we will define a generic drawing, and we will see that the two usesof the term “generic” line up. Question 1.
Is every generalized dissection close to a generic one?
Question 2.
Is every dissection close to a generic one?
These are questions of incidence geometry, and the answers may depend on theunderlying field. The meaning of “close” will be made precise in Section 4.
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GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 9 v Figure 6.
A non-generic dissection, and two generic dissections.3.
Constrained triangulations
A generalized dissection of a square has an associated 2-complex which is home-omorphic to a disk. If we triangulate any non-triangular poofagons, the result leadsto what we call a constrained triangulation . Definition 4. A constrained triangulation T is a pair T = ( T, C ) , where T isan oriented simplicial complex homeomorphic to a disk, and where C = { C i } is a(finite) set of (collinearity) constraints . Vertices on the boundary of T are called corners , and other vertices of T are called interior vertices. Each collinearity con-straint C i is a set of vertices of T of the form Vertices( S i ) where S i is a contiguousset of triangles of T . (This means that there is a connected subgraph of the dualgraph to T whose vertices are the triangles of S i .) We require the sets S i of trianglesto be disjoint, although the constraints C i need not be.A 2-cell of T is called alive or living if there is no constraint containing all ofits vertices.Except in Section 5, a constrained triangulation always has four corners, whichare labeled p , q , r , s in the cyclic order determined by the orientation of T . A note about our usage: much of the modern and classical literature uses theword “triangulation” to distinguish a special type of dissection, namely a simplicialone. However our usage is different. We use the word “dissection,” modified invarious ways, to refer to a concrete (geometric) object, whereas a “triangulation” isan abstract (topological) object. It is helpful to think of a dissection as a drawing of a triangulation; in fact we make this precise in Section 4. (This is how we willdeform a dissection.) So indeed triangulations are always simplicial but a simplicialdissection, which consists of actual triangles in the plane, is not the same thing asa triangulation, which is an abstract simplicial complex.
Proposition 5 ( D ; T ) . Let D be a generalized dissection. There is a constrainedtriangulation T ( D ) whose vertices and living triangles are in 1-1 correspondencewith the vertices and triangles of D .Proof. Triangulate the poofagons of the associated 2-complex arbitrarily and foreach poofagon define a constraint consisting of the vertices of the poofagon, usingthe boundary to determine the cyclic order. Triangulating the poofagons in a different way produces a (slightly) different T satisfying the conclusion of the proposition, and any two such T ’s are related inthis way. In our previous paper [1] we referred to this as a generalized triangulation.
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10 AARON ABRAMS AND JAMIE POMMERSHEIM
A B CD E F p qrs u v
Figure 7.
A (drawing of) an honest triangulation T , with ev-erything labeled. This is Example 2.If C is empty then T is an abstract version of a classical simplicial dissection of asquare. We call this an honest triangulation . All triangles in an honest triangulationare alive. The constrained triangulation associated to any classical dissection withno constrained vertices (i.e., what is classically called a “simplicial dissection”) ishonest.Figure 7 reproduces the honest triangulation T of Example 2. Figure 8 illustratesadditional examples of the form T = ( T, C ), all with the same triangulation T . Example 8 (cf. Example 2) . If a constraint consists of the vertices of a singletriangle, we indicate the constraint by marking the triangle. For instance Figure8a has two constraints, each consisting of three vertices, indicated by the marks inthe triangles.
Example 9 (cf. Examples 2, 4) . Constraints consisting of vertices from multipletriangles are indicated by connecting the marks in the dual triangulation withdotted lines. Figure 8b has a single constraint C consisting of the vertices ofboth marked triangles, i.e., the four vertices { q , s , u, v } . (Unlike with generalizeddissections, these constraints do not come with a cyclic ordering because T is givenso we do not need to ensure that it is a disk.) If we delete the edge uv , turningthe two dead triangles into a quadrilateral, then we obtain the same 2-complex weget by poofing the dissection in Figure 4c; the poofagon is the quadrilateral. Thisfigure shows one possibility for T ( D ), where D is the (generalized) dissection ofFigure 4c. The other is obtained by exchanging the edge uv for the edge qs .Incidentally, without the dotted line this example would be different; there wouldbe two separate constraints that intersect in u and v . This is analyzed in Example14 in Section 5.2. This possibility is why we mark the triangles rather than shadingthem, as we did with poofagons. Example 10 (cf. Example 2) . Figure 8c shows an extreme example of a constrainedtriangulation. Clearly this does not arise as T ( D ) for any generalized dissection D . Example 11 ( ACE again; cf. Examples 2, 6) . Figure 8d is the constrained trian-gulation for the
ACE example, so named because the living triangles are labeled
A, C, E in Figure 7. (Compare with Figure 5.) Here C = { q vu, rs v, sp u } . Thereis no way to realize this as a classical dissection, without killing one of the living Ceci n’est pas une triangulation.
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GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 11 (a)
Two con-straints again. (b)
Here C contains one con-straint, q v s u . (c) Trianglicide. (d)
Our favorite:the
ACE example.
Figure 8.
Some constrained triangulationstriangles. In Figure 5a the generalized dissection D has one upside-down triangle( s uv ) and the same three constraints that are in C (though there, technically, theconstraints are cyclically ordered). The constrained triangulation of Figure 8d is T ( D ). Example 12. If C contains exactly one constraint and this constraint contains nomore than 2 corners, then T is the type of object we studied in [1]. Also there weusually required that the constraint be non-separating .4. The space of drawings
Definitions and theorem.
We are now ready to introduce the space thatallows us to talk about deforming a (generalized) dissection.
Definition 6.
Let T = ( T, C ) be a constrained triangulation. A drawing of T is amap ρ : Vertices( T ) → C such that (1) for each C ∈ C there is a line ℓ C ⊂ C such that ρ ( v ) ∈ ℓ C for each v ∈ C . (2) the images of the corners form a parallelogram in C ; that is, ρ ( p ) + ρ ( r ) = ρ ( q ) + ρ ( s ) ;The space of all drawings, topologized as a subspace of ( C ) Vertices( T ) , is denoted ˙ X ( T ) .A drawing is generic if in addition to the above, we also have (3) the -gon (parallelogram) ( ρ ( p ) , ρ ( q ) , ρ ( r ) , ρ ( s )) is non-degenerate; (4) if { x, y, z } is the vertex set of a living triangle in T then ( ρ ( x ) , ρ ( y ) , ρ ( z )) is a non-degenerate triangle in C . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
12 AARON ABRAMS AND JAMIE POMMERSHEIM v Figure 9.
This drawing is generic, even though three vertices arelined up horizontally.(5) ρ is injective (in particular this guarantees that the lines ℓ C are uniquelydefined); (6) if C, C ′ are distinct constraints with C ∩ C ′ = ∅ then ℓ C = ℓ C ′ .The closure in ˙ X ( T ) (equivalently in ( C ) Vertices( T ) ) of the set of all generic draw-ings of T is denoted X ( T ) .We call T drawable if there exists a generic drawing of T , i.e., if X ( T ) is non-empty. The space ˙ X is evidently an algebraic variety, and as the closure of an opensubset, X consists of a union of components of ˙ X . In this section we give a param-eterization of X in the drawable case. This shows that X is rational and irreducible;it also follows that at most one component of ˙ X ( T ) can contain a generic drawing. Theorem 7.
Let T be drawable. Then X ( T ) is an irreducible rational varietywhich is one of the components of ˙ X ( T ) . Some comments about the definition:(1) Recall that we have defined the term “generic” already for generalized dis-sections. The connection is that if T is a (drawable) constrained triangulation,then every image of a generic drawing of T is a generic generalized dissection, andconversely, every generic generalized dissection D is the image of a generic drawingof the constrained triangulation T ( D ).(2) Note also that this definition of generic is slightly more liberal than the usualconcept of a general position map of points into the plane (subject to (1) and (2)of course). Namely, we allow collections of vertices to be (accidentally) collinear,as long as such syzygies don’t violate condition (6). The dissection D in Figure 9is generic, for example, and it is a generic drawing of T ( D ). Compare with Figure6. (3) Observe that if D is a generic generalized dissection, the slight ambiguity indefining T ( D ) that arises in Lemma 5 disappears in X , and so X ( D ) = X ( T ( D ))is well-defined even though T ( D ) isn’t. Likewise for ˙ X .(4) Examples of drawable triangulations include all honest triangulations ( C = ∅ )as well as any T ( D ) for a generic generalized dissection D . (The latter follows fromitem (3) of Definition 1.)(5) Questions 1 and 2 can be restated more precisely as follows. Question 1’. Is T ( D ) drawable for all (not necessarily generic) generalized dissec-tions D ? RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 13
Figure 10.
A combinatorially reducible T . Question 2’. Is T ( D ) drawable for all (not necessarily generic) dissections D ?Our main interest here is (generalized) dissections, in which context Theorem 7has the following consequence. Corollary 8.
For any generalized dissection D , the deformation space X ( D ) iseither empty or a rational variety that is a single irreducible component of ˙ X ( D ) .Proof. If X ( D ) = ∅ then T ( D ) is drawable and Theorem 7 applies. Combinatorial irreducibility and drawing orders.
We introduce someterminology before proving Theorem 7. Here is a pop quiz: if w, x, y, z are pointsin the plane, and w, x, y are collinear, and x, y, z are collinear, then must w, x, y, z all be collinear? The answer is no. If x = y then x and y determine a unique lineand w and z must be on it. But if x = y then w and z can be anywhere. Example 13 (cf. Examples 2, 4, 9) . Consider the constrained triangulation shownin Figure 10. This has two separate constraints
C, C ′ (the marks are not joinedby a dotted line). Note that it has no generic drawings, because if we label theinterior vertices x and y , then by the pop quiz any drawing ρ either has ρ ( x ) = ρ ( y ) (violating condition (5) of genericity) or ℓ C = ℓ C ′ (violating condition (6) ofgenericity).This leads to the following definition and lemma, the proof of which is no differentin the general case than it is in the above example. Definition 9.
A constrained triangulation T = ( T, C ) is combinatorially irre-ducible if there is at most one vertex in the intersection C ∩ C ′ , for any twoconstraints C, C ′ ∈ C . Lemma 10.
Every drawable T is combinatorially irreducible. Our proof of Theorem 7 gives an explicit rational parameterization of X ( T ) inthe drawable case. The tool we use is called a drawing order .Let T = ( T, C ) be drawable. It can nevertheless be difficult to actually draw T ,if one chooses an unfavorable order in which to draw the vertices. Let ≤ be a totalorder on Vertices( T ), with(1) p ≤ q ≤ s ≤ r ≤ v for all interior v ∈ Vertices . Associated to ≤ there is an integer-valued function v α v on Vertices defined asfollows. Label the vertices other than the corners by v , . . . , v k so that v i ≤ v j iff i ≤ j . Let C ∈ C be a constraint. For each j = 1 , . . . , k define C ≤ j = C ∩ RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
14 AARON ABRAMS AND JAMIE POMMERSHEIM { p , q , s , r , v , . . . , v j } ⊂ Vertices( T ), and say that C is relevant to v j if v j ∈ C and | C ≤ j | ≥
3. Now define α p = α q = α s = 2, α r = 0, and for j = 1 , . . . , k (2) α j = α v j := 2 − { C ∈ C | C is relevant to v j } . We call the total order ≤ a drawing order if (1) holds and also α j ≥ j . Intuitively, we imagine trying to draw the vertices one by one in the orderdetermined by ≤ . When it is time to place v j , the number of available degrees offreedom is (usually) α j . As long as each α j ≥
0, we will produce a drawing.(It is not necessary to require that the corners come first, but it is convenientfor the parameterization that follows.)
Lemma 11.
Every combinatorially irreducible T has a drawing order.Proof. Let T = ( T, C ) be combinatorially irreducible. If Vertices( T ) = { p , q , r , s } then the order p , q , s , r is a drawing order. So we may assume T has interior vertices v , . . . , v k with k > R be the set of interior vertices of T of valence less than 6. This set isnon-empty by an elementary argument about planar graphs, spelled out in Lemma12. For j >
1, let R j be the set of interior vertices of the graph G − ∪ i
Let G be a finite simple graph embedded in the plane such that theexterior face is bounded by the quadrilateral pqrs . Label the other vertices v , . . . , v k and assume k > . Then some v i has valence less than 6.Proof. Let V = k + 4 denote the total number of vertices, let E denote the numberof edges, and let F denote the number of faces determined by the embedding of G in the plane. It is an easy consequence of Euler’s equation V − E + F = 2 that afinite simple planar graph must contain a vertex of valence less than 6. The pointhere is that p , q , r , s cannot be the only vertices with this property. Our proof alsoinvolves nothing more than Euler’s formula.We may assume G is connected, for otherwise we can find the vertex we seek inany connected component not touching the boundary.Suppose for contradiction that each v i has valence at least 6. Then G connectedand k > p , q , r , s is at least 9. Summingvalences we have 2 E = P v Valence( v ) ≥ k + 9 = 6( V −
4) + 9 = 6 V −
15 and since E ∈ Z this implies E ≥ V − E = P f Length( f ) ≥ F + 1 (where Length( f )denotes the number of edges traversed by a path tracing out the boundary of theface f ). RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 15
Now by Euler we have E + 2 = V + F ≤ E + 73 + 2 E −
13 = E + 2 . We must therefore have equality, so in particular all interior faces are triangles,which is the key to the rest of the argument. This implies P v ∈{ p , q , r , s } Valence( v ) ≥
10, whence P v ∈{ p , q , r , s } Valence( v ) = 10 (because otherwise the first inequalityabove would become E ≥ V −
6, a contradiction). This in turn means only two(half-)edges emanate from the corners and go to the interior. Say one emanatesfrom p . Then the other cannot emanate from p or q or s since all interior facesare triangles. Thus the other emanates from r . Moreover these two edges must bethe same edge pr , again because all interior faces are triangles. Now no interiorvertices can be present, meaning k = 0, a contradiction. Proof of theorem.Lemma 13.
Let T be drawable, and let ≤ be a drawing order. There is a rationalmap g ≤ : Y C α v X ( T ) parameterizing X ( T ) . Moreover, if ρ is a generic drawing of T then ρ has a uniquepreimage under g ≤ , and in fact g ≤ is injective in a neighborhood this preimage. Here we interpret C as the singleton { } . Proof.
We define the rational map g : Y v ∈ Vertices( T ) C α v ( C ) Vertices( T ) as follows. Let x = ( x v ) v ∈ Vertices( T ) be coordinates on the domain of g . Thecoordinate functions g v ( x ) of the point g ( x ) are constructed inductively accordingto the chosen drawing order, as follows.Recall that α p = α q = α s = 2 and α r = 0. We start by defining g p ( x ) = x p ∈ C , g q ( x ) = x q ∈ C , g s = x s ∈ C , and g r ( x ) = x q + x s − x p . Then for eachinterior v ∈ Vertices( T ), we assume that the coordinate functions g w ( x ) for vertices w with w < v have been defined. To define g v , we distinguish the three cases: α v = 0 , , . If α v = 2, then x v is a point in C , and we set g v ( x ) = x v .If α v = 1, then x v ∈ C is a number and there is a (unique) constraint that isrelevant to v . We denote by y and z the first two points with respect to ≤ of thisconstraint, and we set g v ( x ) = x v g y ( x ) + (1 − x v ) g z ( x ).Finally if α v = 0 then ( x v = 0 and) there are two constraints C and C ′ rel-evant to v . Let y, z be the first two elements of C (with respect to ≤ ), and let y ′ , z ′ be the first two elements of C ′ . Since y, z, y ′ , z ′ < v , the rational functions g y ( x ) , g z ( x ) , g y ′ ( x ) , g z ′ ( x ) are already defined. Let g v ( x ) be the rational function in g y ( x ) , g z ( x ) , g y ′ ( x ) , g z ′ ( x ) expressing the coordinates of the intersection of the line L through g y ( x ) , g z ( x ) with the line L ′ through g y ′ ( x ) , g z ′ ( x ). This rational func-tion can be computed explicitly, e.g., using Cramer’s rule. There is a denominator.But T is assumed to be drawable, and in a generic drawing ρ the two lines L and L ′ are distinct and non-parallel. Thus this denominator is not identically zero, andso g v ( x ) is indeed a rational function. RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
16 AARON ABRAMS AND JAMIE POMMERSHEIM
We are now finished defining g = g ≤ , and it remains to analyze its image. Notethat the domain of g is irreducible, so the closure of Im( g ) is an irreducible algebraicvariety. We claim that this closure is exactly X .Let ρ be a generic drawing of T . We define parameters x = ( x v ) (in order)such that g ( x ) = ρ . If α v = 2 the parameter is just ρ ( v ). If α v = 1 then by theinjectivity of ρ we know that ρ ( v ) is an affine combination of the first two verticesin the relevant constraint for v , so the parameter x v is uniquely determined. If α v = 0 then the fact that the lines L and L ′ are distinct and meet at the point ρ ( v )means that ρ ( v ) is the (unique and correct) point parameterized by g . Therefore,the image of g contains all generic drawings, so the closure of Im( g ) contains X .Moreover, if g ( x ) = ρ is a generic drawing then there is an open neighborhood U of x = ( x v ) ∈ Q C α v such that g ( x ′ ) is a generic drawing for any x ′ ∈ U . Thisis because conditions (1) and (2) of the definition of drawing are enforced by thedefinition of g , whereas (3), (4), (5), and (6) are open conditions. Thus U mapsinto X . Since U is dense in the domain of g and X is closed, it follows that theclosure of Im( g ) is contained in X , as desired.Finally, we note that for distinct points x ′ , x ′′ ∈ U , if v is the first vertex forwhich x ′ v = x ′′ v then v has different images in C under the maps g ( x ′ ) and g ( x ′′ ).So g is injective on U . Proof of Theorem 7.
The preceding lemma provides the necessary parameterizationof X ( T ). The inverse of g is an algebraic map, so X is indeed rational. Moreover X ⊂ ˙ X , X is irreducible, and X contains an open set of ˙ X , so X must be anirreducible component of ˙ X . Corollary 14.
Let T be drawable and let ≤ be a drawing order. Then P v α v isindependent of choice of ≤ and is equal to the dimension of X ( T ) . Note that the dimension of X agrees with the heuristic count, namely 6 for thecorners plus 2 for each interior vertex minus 1 for each vertex beyond the secondin any constraint.It is also worth noting that the affine group Aff = Aff ( C ) acts on X , and genericdrawings have trivial stabilizers. In particular X ( T ) is topologically a product of C (for translations) and a cone (for scaling) and is therefore contractible (if it isnon-empty). We do not know if the quotient X/ Aff is contractible.4.4.
Home field advantage.
The space ˙ X consists of maps to C . We concludethis section by arguing that from the point of view of drawing pictures, R wouldwork just as well. It may be interesting to study drawing spaces over other fields. Definition 15.
A constrained triangulation T is really drawable if there is ageneric drawing ρ that maps all vertices into R . Such a ρ is called a real drawing .A constrained triangulation T is positively drawable if there is a real drawing ρ such that for any (oriented) triangle ( p, q, r ) of T , the triangle ( ρ ( p ) , ρ ( q ) , ρ ( r )) in R is oriented positively. Such a ρ is called a positive drawing . Here are some remarks about these definitions.(1) There certainly exist constrained triangulations that are positively draw-able. For instance every honest (unconstrained) triangulation is positivelydrawable. Also, T ( D ) is positively drawable for any generic dissection D , RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 17 since D is a positive drawing of T ( D ). Conversely (the image of) anypositive drawing of any T is a generic dissection.(2) There exist (really) drawable constrained triangulations that are not pos-itively drawable. The smallest one is the ACE example, shown in Figure8d.(3) If a constrained triangulation is drawable then it is really drawable. To seethis, find a drawing order and choose real parameters. Almost all choicesare in the domain of the parameterization g , because any denominator thatvanishes at all real points would also vanish at all complex points. Almostall real parameters in the domain of g have image a real drawing.The inflection points of a complex cubic curve can be used to generatea linear system that is drawable but not really, by the Sylvester-Gallaitheorem (see e.g. [2]). Existence of such things also follows from Mn¨evuniversality [8, 10]. However they do not arise from planar triangulations.(4) There exist constrained triangulations that are not drawable, by Lemma10. Slightly less trivially, there are combinatorially irreducible T ’s that arenot drawable. For instance the constraints could force the boundary paral-lelogram to be degenerate. This may be the only obstruction to drawabilityin the combinatorially irreducible case. See also Question 1.5. Musings about ˙ X This is an article about (generalized) dissections. The notion of a constrainedtriangulation allows us to study deformations of generalized dissections, and as wehave already observed, constrained triangulations of the form T ( D ) have certainpleasant properties: they are always combinatorially irreducible, for instance, andif Question 1 has an affirmative answer then they are always drawable too. Thespace X ( D ) is, as we have shown, an irreducible algebraic variety.The collection of constrained triangulations includes many other interesting ob-jects, though, that may be worthy of study on their own merits. We conclude Part1 by highlighting some examples and general questions about their drawing spaces,as well as some parallels with the theory of realizations spaces for oriented matroids.Nothing in this section is central to the paper, although it is not entirely irrelevanteither. The reader who is anxious to get to the area relations can safely proceed toPart 2.5.1. The boundary.
Because of our interest in Monsky’s theorem, we have sofar only discussed constrained triangulations with four corners (see Definition 4),and we have required drawings to realize the boundary as a parallelogram. Someof the issues we want to mention in this section are particular to that case, butmany are not. For the rest of this section we use the notation T D to indicate aconstrained triangulation with arbitrary boundary, whereas T continues to denotea constrained triangulation with four corners.The spaces ˙ X and X (Definition 6) can be defined for arbitrary T D with the ad-justments that condition (2) should be ignored and condition (3) should require theboundary, whatever it is, to be drawn as a non-degenerate polygon. Combinatorialirreducibility (Definition 9) applies to T D without modification.While we are at it we give one more definition. Given T , we have defined botharbitrary drawings and generic drawings (Definition 6). An intermediate type ofdrawing is one which satisfies conditions (1)–(4) of these definitions; we call these RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
18 AARON ABRAMS AND JAMIE POMMERSHEIM
Figure 11.
A combinatorially reducible T and its (two) irre-ducible factors, T ′ and T ′′ . life-preserving , as living triangles of T are required to be drawn non-degenerately.The closure of the life-preserving drawings is denoted ˆ X . Obviously X ⊂ ˆ X ⊂ ˙ X, and like X , the space ˆ X is the closure of an open subset of ˙ X , hence is a union ofcomponents of ˙ X .For T D , we make the same definition, modifying conditions (2) and (3) as wedid earlier to define ˙ X and X .Our feeling is that X ( T ) captures the intuitive idea of deforming a dissection.However we acknowledge that this is to some extent a matter of taste; any of X, ˆ X, ˙ X could reasonably be thought of as a deformation space for T .A notational aside: The ˙ in ˙ X is meant to evoke the constant map, which isan element of ˙ X , while the ˆ in ˆ X resembles a triangle to remind that (most)functions in ˆ X are faithful on the living triangles. The notation X has no decorationbecause it is used the most.In the next subsection we make some conjectures about ˆ X .5.2. Combinatorial reductions and ˆ X . If T D = ( T, C ) is not combinatoriallyirreducible, then it can be decomposed into combinatorially irreducible factors. If u, v ∈ C ∩ C ′ for distinct C, C ′ ∈ C (and distinct u, v ∈ Vertices( T )) then the reduction of T D = ( T, C ) results in two combinatorial factors (or just factors ) T ′ D and T ′′ D , where: • T ′ D = ( T ′ , C ′ ) where T ′ = T and C ′ = C except that C, C ′ have beenreplaced by their union; • T ′′ D = ( T ′′ , C ′′ ) where T ′′ is the result of identifying vertices u, v of T , and C ′′ is adjusted accordingly (removing any resulting constraints of size lessthan 3).For various reasons, the factors resulting from these operations are not alwaysconstrained triangulations, and even if they are, they may not be combinatoriallyirreducible. Moreover T D may have multiple reductions. Nevertheless, recursivelycontinuing this procedure to its conclusion eventually leads to a “factorization” of T D into a collection of combinatorially irreducible factors. Example 14 (cf. Examples 2, 4, 9, 13) . The basic example to keep in mind is the T shown in Figure 11. All these figures have appeared before; this is a parallelogram.There are two constraints in T . In Figure 11 we have equated the factor T ′ witha generic drawing of it. We have also shown T ′′ , which is an honest triangulation. △ X doesn’t typeset very nicely. RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 19
Both T ′ and T ′′ are combinatorially irreducible and drawable; the figure showsgeneric drawings of both. Note that neither of these is considered a generic drawingof T , though for different reasons.In this case ˙ X ( T ′ ) = X ( T ′ ) (though this is not totally obvious) and ˙ X ( T ′′ ) = X ( T ′′ ). Both are irreducible components of ˙ X ( T ), which has no other components.This is a good time to point out that combinatorial irreducibility is not neces-sary for the existence of a drawing order (compare Lemma 11). Recall that froma drawing order ≤ we produce a parameterization g ≤ of a component of ˙ X ( T ). Inthe combinatorially irreducible case any drawing order will yield the same compo-nent, namely X ( T ). On the other hand in the current example, with the interiorvertices labeled u and v , the drawing order pqsr uv yields a parameterization of thecomponent X ( T ′′ ), whereas the drawing order pq uv sr leads to a parameterizationof the other component X ( T ′ ). Conjecture 1. If T D is a constrained triangulation with arbitrary boundary poly-gon, then T D is combinatorially irreducible if and only if ˆ X ( T D ) is an irreduciblevariety. In the case we have focused on for the majority of this paper, i.e., constrainedtriangulations T with the boundary condition, an extra condition is required tomake the analogous conjecture possible. Definition 16.
A constrained triangulation T (of a parallelogram) is toroidallyirreducible if (a) it is combinatorially irreducible and (b) there do not exist con-straints C, C ′ with { p , q } ⊂ C and { r , s } ⊂ C ′ and (c) there do not exist constraints C, C ′ with { q , r } ⊂ C and { p , s } ⊂ C ′ . This is sort of like saying that T is combinatorially irreducible after identifyingopposite edges of the boundary to make T into (a triangulation of) a torus. Wewill not spell out the reduction process but one can imagine that T ′ has a (single)constraint that “wraps around” the torus, and T ′′ is a “constrained triangulationof a segment.”Figure 12 (left) exhibits toroidal reducibility. Here C has two constraints, pq u and rs v (using our usual notation). This is combinatorially irreducible but nottoroidally irreducible.The space ˙ X ( T ) has two components. One component is X ( T ), consisting ofdrawings ρ with non-degenerate boundary pqrs and with u on the line pq and v onthe line rs . One such drawing is shown in Figure 12 (middle); these drawings are(almost all) generic. This component coincides with X ( T ′ ) and ˙ X ( T ′ ). The othercomponent of ˙ X consists entirely of non-generic drawings ρ having ρ ( p ) = ρ ( q )and ρ ( r ) = ρ ( s ); these might be thought of as “dissections of a segment.” Bothcomponents are 8-dimensional (2-dimensional after quotienting by the affine groupaction). Conjecture 2. If T is toroidally irreducible then ˆ X ( T ) is irreducible. Components of ˙ X . We give some more examples to illustrate the differencesbetween X, ˆ X, and ˙ X .For honest triangulations we have X = ˆ X = ˙ X ; this space is non-empty andirreducible and isomorphic to an affine space. This is technically not a drawing order but the point remains.
RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
20 AARON ABRAMS AND JAMIE POMMERSHEIM p = qs = r Figure 12.
A toroidally reducible T (left) and drawings of its twoirreducible factors, T ′ and T ′′ . Figure 13.
Collateral damage.Example 14 (Figure 11) has ∅ = X = ˆ X = ˙ X , and the latter has two compo-nents of the same dimension. This example is combinatorially reducible, hence not(generically) drawable. The components of ˆ X = ˙ X are X ( T ′ ) and X ( T ′′ ) where T ′ and T ′′ are the factors of T . In other words although there is no generic drawing of T , every drawing of T is close to a generic drawing of one of the two combinatoriallyirreducible factors of T .Example 8 (Figure 8a) is drawable, and we have ∅ 6 = X = ˆ X = ˙ X ; again ˙ X hastwo components of the same dimension. This example is not toroidally irreducible.The components of ˆ X = ˙ X are again X ( T ′ ) and X ( T ′′ ) where the two factors areobtained by the toroidal reduction alluded to above. The first of these is also X ( T ).A new example shown in Figure 13 exhibits ∅ = X = ˆ X = ˙ X , and ˙ X has just onecomponent, consisting of drawings with all five vertices on a line. The phenomenonon display here is called collateral damage .At the opposite extreme from the honest case, suppose that T = ( T, C ) where thevertices of each triangle of T form a constraint. Here of course ˆ X ( T ) = X ( T ) = ∅ .The space ˙ X has a component consisting of drawings in which all points arecollinear, but there may also be other components of smaller dimension. (In Ex-ample 10, ˙ X has just one component.) The space ˙ X ( T ) plays a role in our studybecause it is a model for the base locus of the area map Area : X ( T ) Y ( T ) asso-ciated to the honest T . (In fact that base locus is always contained in ˙ X ( T ), thoughthis may be a proper containment.) We analyze this base locus in a forthcomingpaper.We do not know if it is possible to have X = ˆ X = ˙ X . What about in thedrawable case, i.e., is ∅ 6 = X = ˆ X = ˙ X possible?It may be the case that components of ˙ X ( T ) can always be interpreted as X ( T ∗ )for the various combinatorial factors T ∗ of T . Some of the components, thosemaking up ˆ X , are X ( T ∗ ) for the factors that are themselves drawable constrained RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 21 triangulations. If there is only one of these with non-degenerate boundary, it is also X ( T ). However, this picture is merely conjectural. Question 3.
How many components does ˙ X have, and what are their dimensions? In particular we do not know if ˙ X can have components of dimension largerthan dim X or dim ˆ X , when either of these two is non-empty. (In the drawablecase, of course, Corollary 14 gives the dimension of X .) As this may involve subtleissues in incidence geometry, the answer may again (like Question 1) depend on theunderlying field.Recall that the heuristic dimension count is 6 for the corners, 2 for each additionalvertex, and − T D , the heuristic is the same except the boundary contributes 2 m if it has m vertices.) Corollary 14 verifies this for the component X of ˙ X , when the former isnon-empty. It is possible that in the absence of collateral damage this holds for all T , and even T D . As far as we know, though, the dimension could be higher thanthe heuristic indicates, because the constraints could be redundant (in obvious orsubtle ways). Here is an “obvious” example: constraints abc, bcd, acd are equivalentto abcd . Thus these three really only cut the dimension down by 2. This is becausethe third constraint selects a component of the reducible variety determined by thefirst two. The heuristic is too low by 1.A non-obvious redundancy could arise if there were a (non-trivial) incidencetheorem, such as Pappus. Here there are nine points, and the collinearity of eightspecific triples implies the collinearity of a ninth. So, after accounting for the eighthypothesized constraints, further including the ninth lowers the heuristic dimensioncount but doesn’t actually change the variety.Things like this (probably) are what make the dimension of the realization spacealgorithmically intractable for general point/line configurations. (See below.) For-tunately, Pappus’ theorem does not come into play for us, because we only workwith triangulations of a disk whereas the configuration of Pappus’ theorem is non-planar. However there may be other incidence theorems and we do not knowwhether any non-obvious redundancies can arise in the setting of constrained tri-angulations.5.4.
Realization spaces of oriented matroids.
The issues we have mentioned sofar in this section are reminiscent of general questions about realizing configurationsof points and lines. We now focus on this analogy, and we present a few variationson several problems about oriented matroids that are known to be difficult.At points of ˙ X , although certain triangles are required to be degenerate, theothers have no restriction one way or the other. As a result ˙ X contains all constantfunctions, and ˙ X decomposes as a product of C (due to translations) and a cone(due to scaling). In particular ˙ X is contractible. It may be interesting to study thetopology of the quotient of ˙ X by the affine action.By contrast, in the context of point/line configurations (commonly described inthe language of oriented matroids), there are point/line configurations with dis-connected realization space. This is the “isotopy problem” for point/line config-urations, solved in the 1980’s by various people including the Mn¨ev universality If each triple that is a collinearity hypothesis of Pappus’ theorem is made into a triangle,then the resulting 2-complex made of eight triangles does not embed in the plane, because its1-skeleton contains (a subdivision of) the graph K , . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
22 AARON ABRAMS AND JAMIE POMMERSHEIM theorem [8, 10]. This means there are different drawings of points, with the samepre-specified incidence relations, that are not isotopic to each other through suchconfigurations. This is a restricted sense of isotopy, though, as in a realization spaceone is not allowed to introduce degeneracies (even temporarily) along the way. Inthis sense our deformation spaces are fundamentally different.If Conjecture 1 is true, it would suggest that these “planar” systems are signifi-cantly simpler than general systems, just as planar graphs are significantly simplerthan general graphs.Nevertheless we suspect that Question 3 and its relatives are difficult even forconstrained triangulations.It is worth writing down the following (probably intractable but more basic)question.
Question 4.
Given a finite set
Vertices and a finite collection of subsets C of V ,what is the dimension of the space of those maps ρ : V → C such that for each set C ∈ C , the set of points { ρ ( v ) : v ∈ C } lies in a line? The same question can be asked with C replaced by F n for any field F . Wewould be interested to know if there are results about the hardness of this problem.A related question that we know virtually nothing about is the following. Fixa finite simplicial complex T and a number n such that T embeds in R n . Let d be a function on the simplices of T such that d ( σ ) ≤ dim( σ ) for all σ , and also d ( σ ) ≤ d ( τ ) if σ ⊂ τ . What is the nature of the space X of maps ρ : T → R n satisfying dim( ρ ( σ )) = d ( σ ) for all simplices σ ? What about maps satisfyingdim( ρ ( σ )) ≤ d ( σ )? Part Area relations
We now shift gears and begin our study of Monsky’s theorem and the polynomials f and p discussed in the introduction. Our principal contribution is to extendMonsky’s theorem to the deformation spaces X that we defined in Part 1. We thenexplore the consequences of this extension for f and p .Henceforth all constrained triangulations T will be assumed to have squareboundary. 6. Area of a triangle
Let F be a field of characteristic not equal to 2, and let p i = ( x i , y i ) for i = 1 , , F . We define the area of the (ordered) triangle∆ = ( p , p , p ) to beArea(∆) = Area( p p p ) = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Note that if F = R then this is the usual signed area function. We will also usethis definition when F is C or a function field.Note also that regardless of the field, Area( p p p ) = 0 if and only if p , p , p lie on a line in F . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 23 Monsky Theorems If D is a (classical) dissection of the unit square into triangles with areas a , . . . , a n ,then Monsky’s theorem gives a polynomial f with integer coefficients such that(3) 2 f ( a , . . . , a n ) = 1 . In this section ,we use a modification of Monsky’s argument, carried out over thefield of rational functions in the vertex coordinates, to show that f can be chosento depend only on T ( D ) and not on D , meaning that the a i can represent the areasof the triangles in any drawing of T ( D ), including drawings that are not positive.Accordingly, the equation (3) needs to be modified to take into account the totalarea of the drawing; see (4) below. Moreover, because we carry out the argumentin the setting of abstract triangulations, the theorem will apply equally well togeneralized dissections as to dissections, even when the former have no positivedrawings. Equation (4) will also hold for limits of drawings.We give two versions of this argument, the first for honest triangulations inSection 7.2 and the second that incorporates the constraints in Section 7.3. In thepresence of constraints, these results have significant computational benefit overthe approach taken in [1]. We discuss this further in Section 7.4.7.1. Monsky homogenized and deformed.
We have set up our drawing spacesso that the boundary can be mapped to an arbitrary parallelogram, rather thanjust the unit square. We state our generalization of Monsky’s Theorem in a similarspirit. For this purpose, it makes sense to homogenize the equation of Monsky’sTheorem. This is quite easy to do. Take a dissection of the unit square, and takeany f ∈ Z [ A , . . . , A n ] satisfying Monsky’s theorem for this dissection. Note thatthe polynomial σ = A + · · · + A n evaluates to 1 when the areas a i are plugged in.Thus if we homogenize f with respect to the homogenizing variable σ we obtain ahomogeneous polynomial ˆ f satisfying(4) 2 ˆ f ( a , . . . a n ) = σ ( a , . . . , a n ) e , where e is the degree of f . This relation, now homogeneous, has the advantageof being affine invariant; that is, this equation will hold not only for the originaldissection, but also for any affine image of it.We also wish to find relations that are invariant under deformations. The fol-lowing example illustrates this and introduces some notation used in the statementof the generalized Monsky Theorem. Example 15 (cf. Examples 1, 3) . In Example 1, we introduced the dissection D ofFigure 14. If this is the unit square, and the central vertex has coordinates ( x, y ),then the areas are ˜ Z A = y, ˜ Z B = (1 − x ) , ˜ Z C = (1 − y ) , ˜ Z D = x . The tildes overthe Z ’s indicate that the corners have been fixed to those of the unit square. In amoment we will switch from ˜ Z to Z , when we allow the boundary to be an arbitraryparallelogram. In the spirit of finding relations among the areas that are preservedunder deformations, we consider ˜ Z A , ˜ Z B , ˜ Z C , ˜ Z D to be polynomials (or rationalfunctions) living in the field of rational functions in the two coordinate variables x and y . Contrast this with the a i of Monsky’s Theorem, which are real numbers.We seek algebraic relations among the four rational functions ˜ Z A , ˜ Z B , ˜ Z C , ˜ Z D .In this case, finding such relations is not hard. Corresponding to the geometricobservation that the bottom and top triangles add up to half the area of the square, RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
24 AARON ABRAMS AND JAMIE POMMERSHEIM
A BCD
Figure 14 algebraically we have 2( ˜ Z A + ˜ Z C ) = 1, or more homogeneously,2( ˜ Z A + ˜ Z C ) = σ, where σ = ˜ Z A + ˜ Z B + ˜ Z C + ˜ Z D . Thus the polynomial f ( A, B, C, D ) = A + C satisfies(3), provided the boundary is a unit square.Having found a homogeneous relation of the form 2 f = σ e among the ˜ Z ’s,we now observe that the same relation holds even if the boundary is an arbitraryparallelogram. Precisely, consider the rational function field S in the eight variables x v , y v where v is one of the four vertices other than r , and inside S , define x r = x q + x s − x p and y r = y q + y s − y p . Let Z A , Z B , Z C , Z D ∈ S be the homogeneousquadratic polynomials in these eight variables expressing the areas of the triangleswithout fixing the corners. Then by an easy argument invoking affine invariance,the identical relation 2 f = σ e will hold among the Z ’s. This argument is spelledout in detail in Corollary 21.7.2. Honest triangulations.
We now give our first modification of Monsky’s ar-gument, designed for honest triangulations.The clever uses of ultranorms and Sperner’s lemma in the proofs trace directlyto Monsky’s original theorem [9]. The use of Sperner’s lemma built on an earlierapproach due to Thomas [12]. We emphasize that we are adapting those ideas toour current context. We mimic the treatment in Pete Clark’s class notes [3] whichfleshes out some of the steps.We first establish some notation. Let T be a triangulation of a square with k interior vertices and n = 2 k + 2 triangles. Let S be the rational function field Q ( x v , y v ) for v ∈ Vertices( T ) \{ r } . In S , set x r = x q + x s − x p and y r = y q + y s − y p .The field S is the coordinate function field of the space of drawings of T . Theorem 17 (Monsky+) . Let T be a triangulation of a square and let S be thecoordinate function field of its drawing space. For each triangle ∆ j , let Z j ∈ S bethe homogeneous quadratic polynomial in { x v , y v } expressing the area of ∆ j . Define σ ∈ S by X Z j = σ. Then there exists a homogeneous polynomial f T with integer coefficients in n vari-ables such that f T ( Z , . . . , Z n ) = σ e in S , for some non-negative integer e . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 25
Definition 18. If T is a triangulation of a square with n triangles, then anypolynomial f satisfying the conclusion of Theorem Monsky+ is called a Monskypolynomial for T . For example, the polynomials A + C, B + D, and 2 A − B + 2 C − D are all Monskypolynomials for the example of Figure 14. Proof.
Let R be the subring Z [ Z , . . . , Z n ] ⊂ S . Note that σ ∈ R , and we endeavorto show that σ is an element of the ideal p (2) of R . It then follows that somepower σ e may be written as 2 times an integer polynomial in the Z i . Furthermore,this polynomial can be chosen to be homogeneous of degree e since both σ and allthe Z i are homogeneous of the same degree.Thus all is reduced to showing σ ∈ p (2). Assume this is not the case. Then thereis a minimal prime p containing (2) such that σ p (since p (2) is the intersectionof all prime ideals containing (2)). By Krull’s principal ideal theorem p has height1. Let R p be the ring R localized at the prime ideal p and ¯ R be the integral closureof R p . The ideal p ′ = p R p in R p has height 1. Let q be a prime ideal of ¯ R lyingover p ′ . Then q has height 1 in ¯ R [5].By the Mori-Nagata theorem ¯ R is a Krull domain, hence ¯ R q is a discrete val-uation ring containing ¯ R . The valuation on ¯ R q yields a non-Archimedean ultra-norm k · k on the fraction field of ¯ R q (which is also the fraction field of R ). Since Z j ∈ R ⊂ ¯ R q , we have k Z j k ≤ j . In addition 2 ∈ p so 2 ∈ q ¯ R q so k k <
1. Furthermore σ p so σ is a unit in R p , hence in ¯ R , hence k σ k = 1.Extend the ultranorm k · k to S , referring to [4] as necessary.Now, following Monsky, we color each point φ = ( φ x , φ y ) of the plane S withone of the colors A, B, C via the following comparisons: • if k φ x k ≥ k φ y k and k φ x k ≥ k k then φ gets color A ; • else if k φ y k ≥ k k then φ gets color B ; • else φ gets color C .In other words we color A, B, C according to which of φ x , φ y , or 1 has the largestnorm, breaking ties in that order. (See [9].)Monsky proved two lemmas which hold in this context exactly as he provedthem, using the defining properties of the ultranorm, namely k αβ k = k α k k β k and k α + β k ≤ max {k α k , k β k} , with equality if k α k 6 = k β k . We leave the verificationsas exercises. Lemma 19 (Monsky) . The color of φ agrees with the color of φ + ψ for any C -colored ψ . Lemma 20 (Monsky) . Any triangle ∆ whose vertices are colored ABC satisfies k Area(∆) k > . We intend to use this coloring of S × S to induce a coloring of the vertices of T .We do this as follows.Let M : S × S → S × S be the unique affine transformation on S × S taking( x p , y p ) to (0 , x q , y q ) to (1 , x s , y s ) to (0 , M is (cid:12)(cid:12)(cid:12)(cid:12) x q − x p x s − x p y q − y p y s − y p (cid:12)(cid:12)(cid:12)(cid:12) − , which equals the nonzero element σ of S (so in RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
26 AARON ABRAMS AND JAMIE POMMERSHEIM particular M exists). Thus for any triangle ∆ in S × S we haveArea( M ∆) = 1 σ Area(∆)and since k σ k = 1, k Area( M ∆) k = k Area(∆) k . (Note that we are computing Area in the field S .)Now, as promised, the coloring of S × S induces a coloring of the vertices of T by assigning to the vertex v of T the color of the point M ( v x , v y ). This colorsthe corners pqrs with the colors CAAB and is therefore a Sperner coloring, sothere is an
ABC triangle ∆ j . By Monsky’s lemma 2, k Area( M (∆ j )) k >
1. But k Area( M (∆ j )) k = k Area(∆ j ) k = k Z j k ≤
1, a contradiction. When working with the polynomial f , it is useful to exploit the fact that anyparallelogram is affinely equivalent to the unit square, thereby allowing us to removethe x, y variables corresponding to corners. We make now this idea precise.Given a triangulation T , consider the fixed-corner function field ˜ S , defined to bethe rational function field in the 2 k variables ˜ X v , ˜ Y v , where v denotes an interiorvertex. For corner vertices v , define elements ˜ X v , ˜ Y v ∈ ˜ S , by ( ˜ X p , ˜ Y p ) = (0 , X q , ˜ Y q ) = (1 , X r , ˜ Y r ) = (1 , X s , ˜ Y s ) = (0 , Corollary 21.
Let T be a triangulation of a square and let ˜ S be the fixed-cornerfunction field defined above. For each triangle, let ˜ Z i ∈ ˜ S be the (possibly inho-mogeneous) polynomial of degree less than or equal to expressing the area of thistriangle in the k variables ˜ X v , ˜ Y v . Then there exists a homogeneous polynomial f T with integer coefficients in n variables such that (5) 2 f T ( ˜ Z , . . . , ˜ Z n ) = 1 in ˜ S . Furthermore, for f T , we may take any polynomial satisfying the conclusion ofTheorem Monsky+. Conversely, any homogeneous polynomial of degree e satisfyingequation 5 also satisfies the conclusion of Theorem Monsky+.Proof. Let f T be a polynomial obtained from Theorem Monsky+. For all vertices v , including corners, substitute ˜ X v , ˜ Y v for x v , y v . After these substitutions, each Z i becomes ˜ Z i and σ becomes 1. Hence equation (5) is satisfied in ˜ S .Conversely, suppose that f T is any homogeneous polynomial with integer coeffi-cients satisfying Equation (5). Let M be the map defined in the proof of TheoremMonsky+, and for any vertex v , including corners, define ( ˜ x v , ˜ y v ) = M ( x v , y v ). Inequation (5), substituting ˜ x v , ˜ y v for the variables ˜ X v , ˜ Y v for all interior vertices v turns each ˜ Z i corresponding to triangle T i = uvw into12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x u ˜ x v ˜ x w ˜ y u ˜ y v ˜ y w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x u x v x w y u y v y w (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 σ Z i , the first equation following from the fact that M has determinant 1 /σ . Hence withthis substitution, equation (5) becomes2 f ( 1 σ Z , . . . , σ Z n ) = 1 . Finally, by the homogeneity of f , we get the desired 2 f ( Z , . . . , Z n ) = σ e . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 27
Example 16 (cf. Example 2) . Let T be the triangulation shown in Figure 7. Herethere are 6 triangles with two interior vertices u and v , so by Corollary 21 we maywork in the rational function field in the 4 variables x u , y u , x v , y v . The Z ’s aredefined by 2 Z A = y u Z B = x v y u − x u y v − y u + y v Z C = 1 − x v Z D = x u Z E = x u y v − x v y u − x u + x v Z F = 1 − y v This time, there is no linear polynomial f T satisfying the desired conclusion. How-ever the quadratic f T = ( A + C + E ) − AC + 2 DF + ( A + C + E )( B + D + F )is a Monsky polynomial: it has the property that 2 f T ( Z A , . . . , Z F ) = 1.7.3. Constraints.
We next soup up our previous argument a bit more to take intoaccount the constraints C . We assume T = ( T, C ) is drawable and that we have aparameterization g of X ( T ) coming from a drawing order ≤ as in Theorem 13. Wefind that there is again a polynomial f satisfying (4), this time with the areas ofthe living triangles expressed in terms of the parameters w i of the drawing order.Let T = ( T, C ) be a constrained triangulation of a square that is drawable. Let ≤ be any drawing order, let k = P α i , and let g ≤ : C k → X T be the param-eterizing map defined earlier. Denote the coordinates of C k by w , . . . , w k . Let U = Q ( w , . . . , w k ) be the corresponding field of rational functions in k variables.We call U the parameter field of the drawings of T .Note that the number of living triangles is n = k + 2. Theorem 22 (Monsky++) . Let T = ( T, C ) be a constrained triangulation of asquare that is drawable. Fix a drawing order ≤ with corresponding parameter field U . For each living triangle ∆ j ( ≤ j ≤ n ), let W j ∈ U be the rational function inthe w i expressing the area of ∆ j , i.e., W j is the j th coordinate function of the map Area ◦ g ≤ . Let σ = P W j . Then there exists a homogeneous polynomial f T withinteger coefficients in n variables such that f T ( W , . . . , W n ) = σ e in U , for some non-negative integer e .In fact, if f T (note the font change) is the polynomial promised by Monsky+ forthe honest triangulation T , then we may choose f T to be the polynomial obtainedfrom f T by plugging in zeroes for the variables that correspond to the dead trianglesof T . Definition 23. If T is a constrained triangulation of a square with n living tri-angles, then any polynomial f satisfying the conclusion of Theorem Monsky++ iscalled a Monsky polynomial for T .Proof. Fix T and ≤ . Let m be the total number of triangles (alive and dead) in T . Recall n = k + 2 = P α i + 2 is the number of living triangles. We number thetriangles of T so that the first k + 2 are living in T . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
28 AARON ABRAMS AND JAMIE POMMERSHEIM
Apply Monsky+ to the honest T to get f T ( Z , . . . , Z m ) = 12 σ e in the field S = Q ( { x i , y i | ≤ i ≤ m } ). Here as above Z j is the polynomial in the x i , y i expressing the area of ∆ j . Let f T denote the polynomial f T evaluated withall variables corresponding to dead triangles set to zero.We claim that f T ( W , . . . , W k +2 ) = 12 ( W + · · · + W k +2 ) e in U . To see this, specialize to any point ( w i ) in the domain of g ≤ , where theequation is an equality of complex numbers that is true by Monsky+, becausethese coordinates describe a drawing of T . Since the domain of g ≤ is dense in C ( P α ) , the polynomials must be identical and the claim is established. The preceding theorem could also be proved directly, in a manner very similarto the proof of Monsky+, but invoking the map g .These versions of Monsky’s theorem provide the generalizations promised in theintroduction. Corollary 24. If D is a generalized dissection of a parallelogram (cid:3) into triangleswith areas a , . . . , a n , then there is an integer polynomial f in n variables with f ( a , . . . , a n ) = Area( (cid:3) ) e , for some non-negative integer e . Moreover f can bechosen to be invariant under deformation of D .Proof. If the constrained triangulation T ( D ) is drawable then we may apply Mon-sky++ directly to T ( D ). In any case, even if T ( D ) = ( T, C ) is not drawable, weapply Monsky+ to the honest triangulation T , getting the polynomial f T . Thedissection D is the image of a drawing ρ ; it doesn’t matter that ρ is not generic.As the areas of all dead triangles of T ( D ) are zero in D and any deformation of D , the polynomial f T ( D ) obtained from f T by plugging in zeroes for all variablescorresponding to dead triangles in T ( D ) satisfies the conclusion of the corollary. Example 17 ( ACE again, cf. Examples 2, 6, 11) . Let T be the ACE example,i.e., the constrained triangulation shown in Figure 8d; a generic drawing is shown inFigure 5a. Following the idea of Corollary 21, we fix the corners to be the verticesof the unit square. Here there are 3 living triangles called
A, C, E and two interiorvertices u and v . We use the drawing order in which u < v and see that α u = 1while α v = 0. Thus with fixed corners, there is just one parameter w . The relevantcondition for u is sp u , and the relevant conditions for v are rs v and q vu . For anyvalue of w , the drawing g ( w ) places u at (0 , − w ) and places v at ( w w − , W A = 12 (1 − w ) W C = 12(1 − w ) W E = w w − W A W C = 1, so the polynomial f T = 2 AC RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 29 satisfies 2 f T ( W A , W C , W E ) = 1 and is a Monsky polynomial for T . Indeed, inagreement with Theorem Monsky++, this f T equals the polynomial f T from Ex-ample 16 with the variables B, D, F set to 0.7.4.
Computation.
The last assertion of the Monsky++ theorem is that the poly-nomials f T and f T are related in the most natural way possible. (Of course thesepolynomials are not uniquely defined so this statement is not entirely precise.)However in practice, to compute the polynomial f T ( D ) for a generalized dissection D , we do not actually use this relationship. Doing that would require first invokingMonsky+, computing f T for the corresponding honest triangulation, and then ze-roing out a bunch of variables. But, as these are Gr¨obner basis computations whichgrow very quickly in complexity with the number of variables, it is far preferable toperform the calculation without introducing variables that we know we are even-tually going to evaluate to zero. This is the real value of the Monsky++ theorem:it says that we can work directly with the parameters of the dissection, i.e., thecoordinate functions of g , thereby reducing the variables to only those that areactually needed. This is what we just saw in Example 17, where the deformationspace has only one parameter. As a result one can typically compute f reason-ably quickly for a generalized dissection with up to about 10 living triangles, eventhough the corresponding honest triangulation T may have many more trianglesthan this and attempting to compute f T may crash our computers. The left dis-section of Figure 2, for example, has four parameters. There are six triangles andits Monsky polynomial has degree four, so f has at most (cid:0) (cid:1) = 126 monomialsand it is easily computed (in fact it has 104 monomials). The corresponding honesttriangulation T has 10 triangles and a Monsky polynomial of degree six. This one isstill computable in a reasonable amount of time, but it is quite large and unwieldy.8. The area variety
Following [1], we now introduce the machinery necessary to define the polynomial p , starting with the area map.Given three points ( x , y ) , ( x , y ) , ( x , y ) ∈ C , we have defined the area ofthe oriented triangle ∆ with these vertices (in this order) to beArea(∆) = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We also sometimes write Area( p p p ) for the area of the triangle ∆ with vertices p , p , p ∈ C .Note that Area( p p p ) = 0 if and only if p , p , p lie on a (complex) line in C .When ∆ ⊂ R the function Area gives the usual (signed) area.Let T be a fixed drawable constrained triangulation. Let T , . . . , T n be theliving triangles of T . Note each T i inherits an orientation from T . Let Y ( T ) bethe projective space P n − with coordinates [ · · · : A i : · · · ], 1 ≤ i ≤ n . If T = T ( D )then we also denote Y ( T ) by Y ( D ).As T is drawable, the set of generic drawings is open and dense in X ( T ). Wetherefore have a rational mapArea = Area T : X ( T ) Y ( T )given by Area( ρ ) = [ · · · : Area( T i ) : · · · ]. RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
30 AARON ABRAMS AND JAMIE POMMERSHEIM
Definition 25.
Let T be drawable. The area variety V = V ( T ) is the closure in Y ( T ) of Area( X ( T )) . If T = T ( D ) for some generalized dissection D then we alsorefer to V = V ( T ) = V ( D ) as the area variety of D . Note that the affine group Aff ( C ) acts on X and that the area map is equivariantwith respect to this action. (This accounts for not just the translations and scalingwe alluded to after defining X , but also rotations and shears.) Thus since T isdrawable the generic fibers of the area map are at least 6-dimensional (that beingthe size of Aff ).Meanwhile one easily counts that dim( Y ) = n − X − T the area variety V ( T ) has codimension at least 1 in Y ( T ), with equality if and only if generic fibers are exactly 6-dimensional. In otherwords, V is a hypersurface in Y if and only if there is no 1-parameter family ofarea-preserving deformations, other than those contained in an Aff orbit. Definition 26.
A drawable constrained triangulation T is hyper if V ( T ) is ahypersurface in Y ( T ) . Conjecture 3. If D is a generalized dissection then T ( D ) is hyper, i.e., V ( D ) isa hypersurface in Y ( D ) . At least two phenomena can prevent an arbitrary constrained triangulation frombeing hyper, as we described in Section 4 of [1]. However these phenomena do notarise for constrained triangulations of the form T ( D ).All honest triangulations are hyper, and we proved in [1] that if T has only oneconstraint and it is non-separating then T is hyper. That proof can be extendedsomewhat. In a forthcoming paper we further enlarge the set of T ( D )’s that weknow to be hyper.If T is hyper then there is a unique (up to scaling) non-zero polynomial p = p T that vanishes on V . The polynomial p is irreducible because X , and therefore V ,is an irreducible variety. Also p has rational coefficients (because the coordinatefunctions of Area do) so p can be normalized to have integer coordinates with nocommon factor. We assume this has been done; the polynomial p T is now well-defined up to sign for any hyper T . Definition 27.
We call p and − p the area polynomials for T . We remark that the area polynomials are computable, using Gr¨obner basis tech-niques, but that these computations quickly become intractable as the triangulationgrows. 9.
Mod 2
Let T be a constrained triangulation that is hyper (hence drawable). We thushave Monsky polynomials f and an area polynomial p , both homogeneous elementsof the polynomial ring Z [ A , . . . , A n ], with variables A i corresponding to the livingtriangles of T . Letting σ = P A i , the equations 2 f = σ e and p = 0 hold on thevariety V ( T ). The existence of p and f with these properties is enough to revealall of the coefficients of p modulo 2. In each of those scenarios there is a subset of the variables that sums to zero and as far as weknow the area variety is still a hypersurface in a smaller projective space than Y . So, it is possiblethat a slight modification of Conjecture 3 holds for all T . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 31
Theorem 28 (Mod 2 theorem) . If T is hyper then its area polynomial p satisfies p ≡ σ d mod 2 , where d = deg p .Proof. Since V ( T ) is the zero set of the irreducible polynomial p , any polynomialthat vanishes on V ( T ) is a multiple of p . Thus p | f − σ e . These are integerpolynomials and the divisibility occurs in Q [ A , . . . , A n ], so there is a polynomial q in this ring with p · q = 2 f − σ e . By Gauss’s Lemma, q must have integer coefficients.Therefore we may reduce the coefficients mod 2, and using [ · ] for the reduction, wehave [ p ][ q ] = [ σ e ]. Since Z / Z [ A , . . . , A n ] is a unique factorization domain, weconclude that [ p ] = [ σ d ] for some d = deg p ≤ e . Corollary 29.
Let T be hyper, and suppose the area polynomial p has degree d .Then all leading terms A di occur with odd (hence non-zero) coefficient. In a forthcoming paper we show that the leading coefficients are all equal up tosign. We suspect these coefficients are all ±
1, as we discuss in Section 11.10.
Canonical Monsky polynomials
Let T be hyper, let p be its area polynomial, and suppose p has degree d . Recallthat p is only well-defined up to sign; we suppose we have chosen one of the twopossibilities.Using p , we now single out a particular f satisfying Monsky++. By Theorem 28we have σ d + p = 2 f for some polynomial f ∈ Z [ A , ..., A n ], homogeneous of degree d = deg p . Note that f satisfies the conclusion of Monsky++, since 2 f − σ d = p vanishes on V . This shows that we may choose f = 12 ( σ d + p )in Theorem Monsky++. This shows in addition that we may choose a Monskypolynomial with the same degree as p (and no lower).If we begin with − p instead of p , we end up with˜ f = 12 ( σ d − p ) = f − p instead. The pair { f, ˜ f } is therefore a canonically defined pair of Monsky polyno-mials, both of minimal degree. Definition 30.
For T hyper with area polynomial ± p of degree d , the canonicalMonsky polynomials are { f, ˜ f } where f = ( σ d + p ) and ˜ f = ( σ d − p ) . Proposition 31.
For T hyper with area polynomials ± p of degree d , the polynomial f is a Monsky polynomial for T if and only if f = ( σ e + pq ) for some integer e ≥ d and for some polynomial q ≡ σ e − d mod 2 .In particular, the minimal degree Monsky polynomials are exactly { f + mp } = { ˜ f + mp } , where f, ˜ f are the canonical Monsky polynomials and where m is anyinteger.Proof. For polynomials f ∈ Z [ A , ..., A n ], the condition that f be a Monsky poly-nomial is equivalent to the condition that p | f − σ e for some e , which in turn isequivalent to the condition that f = ( σ e + pq ) for some q ≡ σ e − d mod 2. Thesecond assertion follows by considering e = d . RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
32 AARON ABRAMS AND JAMIE POMMERSHEIM
The canonical Monsky polynomials f, ˜ f satisfy f + ˜ f = σ d (6) f − ˜ f = p (7)(where interchanging f and ˜ f corresponds to interchanging p and − p ). Example 18 (cf. Examples 1, 3, 15) . We saw in Example 15 that for the trian-gulation in Figure 4a, A + C is a Monsky polynomial. The area polynomial is p = A − B + C − D and so the canonical Monsky polynomials are f = B + D and˜ f = A + C . We have 2 f = 2 ˜ f = σ in the field U and on the variety V . OtherMonsky polynomials may be obtained by adding a multiple of p to f . For example, p + f = 2 A − B + 2 C − D is a Monsky polynomial. Example 19 (cf. Example 2, 16) . For the honest triangulation T of Figure 7 wechoose the area polynomial p = ( A + C + E ) − AC − ( B + D + F ) + 4 DF . (Seealso [1], where this is worked out in detail.) This is irreducible and it vanishes on V . The canonical Monsky polynomials are f = 12 ( σ + p ) = ( A + C + E ) − AC + 2 DF + ( A + C + E )( B + D + F )˜ f = 12 ( σ − p ) = ( B + D + F ) − DF + 2 AC + ( A + C + E )( B + D + F ) . This is how we found the polynomial f T given in Example 2 from the introductionand again in Example 16 in Section 7.2.Notice in this example that there is another way to obtain f and ˜ f from p . If wewrite p = p + − p − as the difference of two polynomials with non-negative coefficientsand no common terms, then we have p + = p − on V , where p + = A + C + E + 2 AE + 2 CE + 2 DF and p − = B + D + F + 2 BD + 2 BF + 2 AC.
We see that each of these coefficients is less than or equal to the correspondingcoefficient in the expansion of σ . The terms of this expansion that do not occurin p + or p − are 2( A + C + E )( B + D + F ). Thus we can “make up the difference”by adding t = ( A + C + E )( B + D + F ) to both sides, giving p + + t = p − + t on V , and ( p + + t ) + ( p − + t ) = σ d (8) ( p + + t ) − ( p − + t ) = p. (9)This means we have found two polynomials whose sum is σ d and whose differencevanishes on V ; therefore each is half of σ d and they are Monsky polynomials.Comparing with (6), we see that in fact f = p + + t and˜ f = p − + t. Positivity
Something happened in the last derivation that may not always work. When wewrote p + and p − , we observed that all the terms have coefficients that are “small,”in the sense that none is larger than the corresponding coefficient of σ . As a direct RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 33 consequence, all coefficients of t , hence also of both f and ˜ f , turned out to benon-negative.This phenomenon is not essential to the procedure; if it fails one can still define t satisfying (8) and proceed to determine f and ˜ f . In that case t and at least oneof f, ˜ f would have some negative coefficients. For honest triangulations, however,we have never observed this. Definition 32.
A polynomial is called positive if all its coefficients are non-negative.
Conjecture 4 (Positivity) . The canonical Monsky polynomials of every honesttriangulation are positive.
We can restate this conjecture in terms of the area polynomial as follows.
Definition 33.
A homogeneous polynomial p of degree d is called small if eachcoefficient of p has absolute value less than or equal to the corresponding multinomialcoefficient; that is, if both σ d − p and σ d + p are positive. Because of the relationships 2 f = σ d + p and 2 ˜ f = σ d − p , the positivity conjectureis equivalent to saying that the area polynomial of an honest triangulation is small.At the end of the previous section we mentioned our suspicion that the leadingterms A di of the area polynomial p all have coefficient ±
1. We point out now thatthis is a special case of the positivity conjecture.We conclude this section with some further remarks about this conjecture. Weframe these remarks in terms of the area polynomial p , rather than f , because wehave more techniques for computing and working with p .Observe that smallness is preserved under products: if p and ¯ p are two smallpolynomials, then p ¯ p is also small.Observe also that smallness is a local condition on the polynomial p , in thesense that its failure is always witnessed by (at least) one individual monomial. Inother words p is small if and only if each monomial of p is small, even though agiven monomial may not include all the variables in the polynomial p . For instanceany polynomial containing the term, say, − ABCD fails to be small, because thedegree is 4 and | − | is larger than the coefficient of ABCD in σ , which is 24regardless of how many variables there are in σ .With these observations, and using the methods of [1], we can show that thepositivity conjecture holds for the infinite family T n of honest triangulations shownin Figure 15. We call this the “diagonal case.”Note that T and T have already made numerous appearances in this paperunder the pseudonyms Example 1 and Example 2. Proposition 34.
For each n , the diagonal case T n has positive Monsky polynomial.Proof. As it is honest, T n is hyper, and we denote its area polynomial by p n . In[1] we gave an explicit expression for p n , but it is difficult to tell directly from thatexpression that p n is small. However we do know that the degree of p n is exactly n . If we focus on any particular monomial of p n , then at least one subscript from { , . . . , n + 1 } , call it j , does not occur in the variables of this monomial. If wekill triangles A j and B j the resulting polynomial p n | A j = B j =0 factors into a linearfactor with coefficients ± n − p n − (but with subscripts at least j shifted up by one). The linear factor is small, and RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
34 AARON ABRAMS AND JAMIE POMMERSHEIM A A A A n +1 B B B B n +1 p p p p n Figure 15.
The diagonal case T n .inductively so is p n − , so their product is also small. The monomial from p n onwhich we focused is one term of this product, and so this chosen monomial satisfiessmallness. Since our choice of monomial was arbitrary, it follows that p n is small,as desired. Example 20 ( ACE again, cf. Examples 2, 6, 11, 17) . To see some non-honestexamples, one can start with the honest triangulation T and plug in zeroes. Forinstance consider the ACE example. Monsky++ implies that we may take the f and ˜ f computed already for the honest case and plug in B = D = F = 0, giving f = A + C + E + 2 AE + 2 CE ˜ f = 2 AC.
The area polynomial p = ( A + C + E ) − AC is likewise obtained by plugging in B = D = F = 0 from Example 16. We check f + ˜ f = ( A + C + E ) and f − ˜ f = p .There is a reason that we assume honesty in the positivity conjecture. The nextexample shows a classical dissection D whose T ( D ) has an area polynomial that isnot small. Example 21 (Failure of positivity) . We return to the second figure in this paper,reproduced in Figure 16. The constrained triangulation T = T ( D ) = ( T, C ) has tentriangles and four constraints; there are six living triangles. The area polynomial p T for the honest triangulation T has degree six, and is small. However, the areapolynomial p T for the constrained triangulation has degree four and has a total of70 terms, one of which is − ABCD , where
A, B, C, D denote the areas of the fourtriangles that touch the corners. Therefore p T is not small. Of course, this termdoes not occur in the degree six polynomial p T .Curiously, of the 70 terms of p T , only one violates smallness. Likewise, of the104 terms of f and the 122 terms of ˜ f , just one of them has a negative coefficient,namely − ABCD . 12.
Equidissections
Monsky’s original equidissection theorem leaves open the question of which dis-sections can be deformed to be equi-areal. (It is easy to see that for each even n ,there exist such dissections with n triangles.) Observe that if D is a dissection with n triangles and p (1 , , . . . , = 0, or equivalently the sum of the coefficients of f RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT --
GENERALIZED DISSECTIONS AND MONSKY’S THEOREM 35
Figure 16. A T whose Monsky polynomial is not positive. Figure 17.
This dissection cannot be deformed to an equidissection.not equal to n d /
2, then D cannot be deformed to an equidissection without killingtriangles. This is the case, for instance, with the dissection shown in Figure 17:there are 8 triangles, and the polynomial p has degree 3, so plugging in all 1’s to σ d gives the value 8 = 512. However plugging in all 1’s in f and ˜ f gives the values260 and 252, which are not equal (and p (1 , . . . ,
1) is the difference, ± /
8. In this case,this is also easily proved using elementary Euclidean geometry.
Question 5.
Given a dissection or generalized dissection D , can one predict fromthe combinatorics of T ( D ) whether or not D can be deformed to an equidissection? acknowledgements The authors thank the Budapest Semesters in Mathematics program for sup-porting us in 2019 as Director’s Mathematicians in Residence. The first authoracknowledges additional support from the Simons Foundation (grant
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