Tropical types and associated cellular resolutions

0.34 MB·English

by  Anton Dochtermann

TROPICAL TYPES AND
ASSOCIATED CELLULAR RESOLUTIONS
ANTONDOCHTERMANN,MICHAEL JOSWIG, AND RAMANSANYAL
0
1 Abstract. AnarrangementoffinitelymanytropicalhyperplanesinthetropicaltorusTd−1
0 leadstoanotionof‘type’dataforpointsinTd−1,withtheunderlyingunlabeledarrangement
2 givingriseto‘coarsetype’. ItisshownthatthedecompositionofTd−1inducedbytypesgives
risetominimal cocellular resolutions ofcertain associated monomial ideals. ViatheCayley
n
a trick from geometric combinatorics this also yields cellular resolutions supported on mixed
J subdivisions of dilated simplices, extending previously known constructions. Moreover, the

8 methods developed lead to an algebraic algorithm for computing the facial structure of
1 arbitrary tropical complexes from point data.

]
O
C 1. Introduction
.
h
The study of convexity over the tropical semiring has been an area of active research in
t
a recentyears. Fundamentalpropertiesoftropicalconvexity, inparticularfromacombinatorial
m
perspective, were established by Develin and Sturmfels in [7]. There the notion of a tropical
[
polytope was defined as the tropical convex hull of a finite set of points in the tropical torus

2 Td−1. Fixing the set of generating points yields a decomposition of the tropical polytope
v called the tropical complex, and in [7] it was shown that the collection of such complexes are
7
in bijection with the regular subdivisions of a product of simplices. The origin of tropical
3
2 convexity can be traced back to the study of ‘max-plus linear algebra’; see [18, 5] and the
0 references therein for a proper account.
.
1 A tropical complex can be realized as the subcomplex of bounded cells of the polyhedral
0 complex arising from an arrangement of tropical hyperplanes, and it is this perspective that
0
we adopt in this paper. We study combinatorial properties of arrangements of tropical
1
: hyperplanes, and in particular their relation to algebraic properties of associated monomial
v
ideals. A tropical hyperplane in Td−1 is defined as the locus of ‘tropical vanishing’ of a linear
i
X form, and can be regarded as a fan polar to a (d−1)-dimensional simplex. In this way each
r tropical hyperplane divides the ambient space Td−1 into d sectors. Given an arrangement A
a
of tropical hyperplanes and a point p ∈ Td−1, one can record the position of p with respect
to each sector of each hyperplane. This tropical analog of the covector data of an oriented
matroid is called the type data, and the combinatorial approach to tropical convexity taken
in [7] is based on this concept. Here we consider a coarsening of the type data (which we call
coarse type) arising from an arrangement A of tropical hyperplanes, amounting to neglecting
the labels on the individual hyperplanes.
The connection between tropical polytopes/complexes and resolutions of monomial ideals
was first exploited by Block and Yu in [4]. There the authors associate a monomial ideal to
Research of Dochtermann supported by an Alexander von Humboldtpostdoctoral fellowship.
Research of Joswig supported byDeutscheForschungsgemeinschaft, DFGResearch Unit “Polyhedral Sur-
faces”.
Research of Sanyal supported by the Konrad-Zuse-Zentrum fu¨r Informationstechnik (ZIB) and a Miller
Research Fellowship.
1
2 ANTONDOCHTERMANN,MICHAELJOSWIG, ANDRAMANSANYAL
a tropical polytope with generators in general position, and use algebraic properties of its
minimal resolution to determine the facial structure of the bounded subcomplex. Further
progress along these lines was made in [8]. The primary tool employed in this context is that
of a cellular resolution of a monomial ideal I, where the i-th syzygies of I are encoded by the
i-dimensionalfacesofapolyhedralcomplex(seeSection 3). Theideals from[4]aresquarefree
monomials idealsgenerated bythecotype data, i.e.thecomplements ofthetropicalcovectors,
arising from the associated arrangement of hyperplanes, and hence can be seen as a tropical
analog of the (oriented) matroid ideals studied by Novik, Postnikov and Sturmfels in [22].
In this paper we study the polyhedral complex C (and its bounded subcomplex B )
A A
induced by the type data of an arrangement A of n hyperplanes in Td−1. Both complexes are
naturally labeled by fine and coarse type and cotype data, and we show how the resulting
labeledcomplexessupportminimal(co)cellular resolutionsofassociatedmonomialideals. We
pay special attention to labels given by coarse type. For instance, we show that C supports
A
a minimal cocellular resolution of the ideal It(A) generated by monomials corresponding to
the set of all coarse types. The proof involves a consideration of the topology of certain
subsets of C as well as the combinatorial properties of the coarse type labelings. When
A
the arrangement A is sufficiently generic we show that the resulting ideal is always given by
hx1,...,xdin, the n-th power of the maximal homogeneous ideal; in general, It(A) is some
Artinian subideal. Our results in this area are all independent of the characteristic of the
coefficient field.
Via the connection to products of simplices and the Cayley trick we interpret these results
in the context of mixed subdivisions of dilated simplices. In particular we obtain a minimal
cellular resolution of It(A) supported on a subcomplex of the dilated simplex n∆d−1. One
other direct consequence is that any regular fine mixed subdivision of n∆ supports a
d−1
minimal resolution of hx ,...,x in. This extends a result of Sinefakopoulos from [29] where
1 d
a particular subdivision is considered (although much less explicitly), and also complements
a construction of Engstr¨om and the first author from [9] where such complexes are applied
to resolutions of hypergraph edge ideals. The duality between tropical complexes and mixed
subdivisions of dilated simplices was established in [7], and we show how this extends to the
algebraic level in terms of Alexander duality of our resolutions of the coarse type and cotype
ideals.
Finally, we show how these algebraic results lead to observations regarding the combina-
toricsoftropicalpolytopes/complexesandmixedsubdivisionsofdilatedsimplices. Weobtain
a formula for the f-vector of the boundedsubcomplex of an arbitrary tropical hyperplane ar-
rangement in terms of the Betti numbers of the associated coarse type ideal. The uniqueness
of minimalresolutions also implies that for any sufficiently generic arrangement A, themulti-
set of coarse types is independent of the arrangement. Furthermore, we present an algorithm
for determining the incidence face structureof a tropical complex from the coordinates of the
generic set of vertices, utilizing the fact that such a complex supports a minimal resolution
of the square-free monomial cotype ideal. This approach was first introduced by Block and
Yu in [4] for the case of sufficiently generic arrangements, and we extend the algorithm to
the general case.
The rest of the paper is organized as follows. In Section 2 we review the basic notions of
tropical convexity including tropical hyperplanes and type data, and discuss the polytopal
complex that arises from an arrangement of tropical hyperplanes. We introduce the notion
of coarse type and establish some of the basic properties that will be used later in the paper.
In Section 3 weintroduce themonomial ideals that arise from an arrangement of hyperplanes
and show that the polytopal complexes labeled by fineandcoarse (co)type supportcocellular
TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 3
(and cellular) resolutions. In Section 4 we recall the construction of the Cayley trick and use
this to interpret our results in terms of mixed subdivisions of dilated simplices. In Section 5
we discuss some examples of our construction, in particular the case of the (generic) staircase
triangulation (recovering a result of [29]) as well as a family of non-generic arrangements
correspondingto tropical hypersimplices. In Section 6 we show our results give rise to certain
consequences for the combinatorics (e.g., the f-vector) of the boundedsubcomplexes of trop-
ical hyperplane arrangements, and also describe an algorithm for determining the entire face
poset from the coordinates of the arrangement. This strengthens a result from [4]. Finally,
we end in Section 7 with some concluding remarks and open questions.
We are grateful to Kirsten Schmitz for very careful proof reading.
2. Tropical convexity and coarse types
In order to fix our notation we begin in this section with a brief review of the foundations of
tropical convexity as layed out by Develin and Sturmfels in [7]. We then define ‘coarse types’
and establish some combinatorial and topological results regarding the type decomposition
of the tropical torus induced by a finite set of points. While some of these observations
may be worthwhile in their own right, their main interest for us will be their applications to
subsequent constructions of (co)cellular resolutions.
2.1. Tropical convexity and tropical hyperplane arrangements. Tropical convexity
is concerned with linear algebra over the tropical semi-ring (R,⊕,⊙), where
x⊕y := min(x,y) and x⊙y := x+y.
We will sometimes replace the operation min with max, and although the two resulting
semi-rings are isomorphic via
−max(x,y) = min(−x,−y),
it will be useful for us to consider both structures on the set R simultaneously. To avoid
confusion we will therefore use the terms min-tropical semi-ring and max-tropical semi-ring,
respectively. Componentwise tropical addition and tropical scalar multiplication turn Rd
into a semi-module. The tropical torus Td−1 is the quotient of Euclidean space Rd by the
linear subspace R , where ∈ Rd is the all-ones vector. By interpreting this quotient in the
category oftopolo1gical spac1es,Td−1 inheritsanaturaltopologywhichishomeomorphictothe
usualtopology on Rd−1. A set S ⊂ Td−1 is tropically convex if it contains (λ⊙x)⊕(µ⊙y) for
all x,y ∈ S and λ,µ ∈ R. For an arbitrary set S ⊂ Td−1 the tropical convex hull tconv(S) is
definedas the smallest tropically convex set containing S. If the set S is finite, then tconv(S)
is called a tropical polytope. In this paper, tropical convexity and related notions will be
studied with respect to both min and max, and hence we will also talk about max-tropically
convex sets and the like.
The tropical hyperplane with apex −a ∈ Td−1 is the set
H(−a) := p ∈ Td−1 : (a ⊙p )⊕(a ⊙p )⊕···⊕(a ⊙p ) is attained at least twice .
1 1 2 2 d d
n o
That is, a tropical hyperplane is the tropical vanishing locus of a polynomial homogeneous
of degree 1 with real coefficients. If we want to explicitly distinguish between the min-
and max-versions we write Hmin(−a) and Hmax(−a), respectively. Any two min-tropical
(resp. max-tropical) hyperplanes are related by an ordinary translation, and hence a tropical
hyperplane is completely determined by its apex. Furthermore, the min-tropical hyperplane
with apex 0 is a mirror image of the max-tropical hyperplane with apex 0 under the map
4 ANTONDOCHTERMANN,MICHAELJOSWIG, ANDRAMANSANYAL
x 7→ −x. ThecomplementofanytropicalhyperplaneinTd−1 consistsofpreciselydconnected
components, its open sectors. Each open sector is convex, in both the tropical and ordinary
sense. The k-th (closed) sector of the max-tropical hyperplane with apex a is the set
Smax(a) := p ∈ Td−1 : a −p ≤ a −p for all i∈ [d] .
k k k i i
n o
Similarly we have
Smin(a) := p ∈ Td−1 : a −p ≥a −p for all i ∈ [d]
k k k i i
n o
for the min-version. Notice that x ∈ Smax(y) if and only if y ∈ Smin(x). Each closed sector is
k k
thetopologicalclosureofanopensector. Againtheclosedsectorsaretropicallyandordinarily
convex. A sequence of points V = (v ,v ,...,v ) in Td−1 gives rise to the arrangement
1 2 n
A(V) := Hmax(v ),Hmax(v ),...,Hmax(v )
1 2 n
(cid:0) (cid:1)
of n labeled max-tropical hyperplanes. The position of points in Td−1 relative to each hyper-
plane in the arrangement furnishes combinatorial data and leads to the following definition.
Definition 2.1 (Fine type). Let A= A(V) be the arrangement of max-tropical hyperplanes
given by V = (v ,v ,...,v ) in Td−1. The fine type (or sometimes simply type) of a point
1 2 n
p ∈ Td−1 with respect to A is the table T (p)∈ {0,1}n×d with
A
T (p) = 1 if and only if p ∈ Smax(v )
A ik k i
for i ∈ [n] and k ∈ [d]. The fine cotype T (p) ∈ {0,1}n×d is defined as the complementary
A
matrix, that is,
T (p) = 1 if and only if T (p) = 0.
A ik A ik
Let us now fix a sequence of points V and the corresponding arrangement A = A(V) in
Td−1. We write T(p) instead of T (p) when no confusion arises. For a fixed type T = T(p),
A
the set
C◦ := q ∈ Td−1 : T(q) = T
T
n o
of points in Td−1 with that type is a relatively open subset of Td−1 = Rd−1; called the
relatively open cell of type T. The set C◦ as well as its topological closure C are tropically
T T
and ordinarily convex. The set of all relatively open cells C◦ = C◦(V) partitions the tropical
torus Td−1. Moreover, C , the closed cell of type T, is the intersection of finitely many closed
T
sectors. Since each closed sector is the intersection of finitely many polyhedral cones, this
implies that C , provided it is bounded, is a polytope in both the classical and the tropical
T
sense; in [14] these were called polytropes. Thecollection of all closed cells yields a polyhedral
subdivision C = C(V) of Td−1. The collection of types with the componentwise order is
anti-isomorphic to the face lattice of C(V), that is, T (D) ≤ T (C) whenever C ⊆ D are
A A
closed cells of C(V). The cells which are bounded form the bounded subcomplex B = B(V).
Following Ardila and Develin [1], a fine type should be thought of as the tropical equivalent
of a covector in the setting of tropical oriented matroids, with the cells of maximal dimension
playing the role of the topes. The following result highlights the relation of min-tropical
polytopes and max-tropical hyperplane arrangements.
Theorem 2.2 ([7, Thm. 15 and Prop. 16]). The min-tropical polytope tconv(V) is the union
of cells in the bounded subcomplex B(V) of the cell decomposition of Td−1 induced by the
max-tropical hyperplane arrangement A(V).
TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 5
The polytopal complex B(V) induced by types is a subdivision of the tropical polytope
tconv(P) called the tropical complex generated by V. The points V are in tropically general
position if the combinatorial type of C(V) (or equivalently B(V)) is invariant under small
perturbations. In Section 4, we will elaborate on the important connection between trop-
ical hyperplane arrangements and mixed subdivisions of dilated simplices. In light of this
connection, the points V are in general position if the associated mixed subdivision is fine.
At this point we introduce our running example, borrowed from [4, Ex. 10].
Example 2.3. The points v = (0,3,6), v = (0,5,2), v = (0,0,1), and v = (1,5,0)
1 2 3 4
give rise to the max-tropical hyperplane arrangement shown in Figure 1. It decomposes the
tropical torus T2 into 15 two-dimensional cells, three of which are bounded. Note that there
is precisely one bounded cell of dimension one (incident with the 0-cell v ) which is maximal
1
with respect to inclusion. This shows that the polytopal complex B(V) need not be pure.
Moreover, one can check that the dual regular subdivision of ∆ ×∆ is a triangulation, and
3 2
hence these four points are sufficiently generic.
2.2. Coarse types. As we have seen, the fine type records the position of a point relative
to a labeled tropical hyperplane arrangement. Neglecting the labels on the hyperplanes leads
to the following coarsening of the type information.
Definition 2.4 (Coarse type). Let A = A(V) be an arrangement of n max-tropical hy-
perplanes in Td−1. The coarse type of a point p ∈ Td−1 with respect to A is given by
t (p) = (t ,t ,...,t ) ∈ Nd with
A 1 2 d
n
t = T (p)
k A ik
Xi=1
for k ∈[d].
The coarse type entry t records for how many hyperplanes in A the point p lies in the
k
k-th closed sector. Again we will write t(p) when no confusion can arise. By definition t
is constant when restricted to a relatively open cell C◦. The induced map C◦ → Nd is also
T
denoted by t.
Objects in tropical geometry can be lifted to objects in classical algebraic geometry by
considering fields of formal Puiseux series and their valuations. Following this path leads
to another interpretation of the coarse types: For v ∈ Td−1 the max-tropical hyperplane
i
Hmax(v ) is the tropicalization tropmax(h ) of a hyperplane given as the vanishing of a ho-
i i
mogeneous linear form h ∈ C{{z}}[x ,x ,...,x ] defined over the field of Puiseux series.
i 1 2 d
Let
(1) h = h ·h ···h
1 2 n
be the product of these n linear forms, so that h is a homogeneous polynomial of degree n in
d indeterminates. Notice that there is no canonical choice for the polynomials h and hence
i
not for h. In the computation below we choose hi to be z−vi1x1+z−vi2x2+···+z−vidxd.
Remark 2.5. Note that the classical Puiseux series have rational exponents, and therefore
the valuation map takes rational values only. For this reason the tropical hypersurface of
a tropical polynomial is often defined as the topological closure of the vanishing locus of
a tropical polynomial; e.g., see Einsiedler, Kapranov, and Lind [11]. By extending C{{z}}
to a field of generalized Puiseux series one can directly deal with real exponents, as the
corresponding valuation map is onto the reals; for a construction see Markwig [19]. In
6 ANTONDOCHTERMANN,MICHAELJOSWIG, ANDRAMANSANYAL
7
004 013
v1
6
5
4
103 022
3
112
v2
2
202
v3 031
1
301 211 121
0
-1
v4
400 310 220 130 040
-2
-1 0 1 2 3 4 5 6
Figure 1. Coarsely labeled type decomposition of T2 induced by four max-
tropical lines. Bounded cells are shaded.
the sequel we will make use of such a field of generalized Puiseux series, and we denote it
by C{{z}}gen.
Proposition 2.6. The tropical hypersurface defined by tropmax(h) is the union of the max-
tropical hyperplanes in A.
The tropical hypersurface defined by tropmax(h) is the orthogonal projection of the co-
dimension-2-skeleton of an unbounded ordinary convex polyhedron P in Rd whose facets
A
correspond to monomials of the polynomial h; see [24, Thm. 3.3].
Proposition 2.7. Let p ∈ Td−1 \A be a generic point. Then its coarse type t (p) is the
A
exponent of the monomial in h which defines the unique facet of P above p.
A
Proof. Since p is generic, it is contained in a unique sector Smax(v ) with respect to the
τi i
hyperplane with apex v . Hence, p satisfies the strict inequalities
i
(2) v −p < v −p
i,τi τi i,j j
for all i ∈ [n] and j ∈ [d]\τ . Equivalently,
i
n
p −v
τi i,τi
Xi=1
is the evaluation of the tropical polynomial tropmax(h) and the corresponding monomial in h
atwhichtheuniquemaximumisattainedisxτ1xτ2···xτn withcoefficientz−v1,τ1−v2,τ2−···−vn,τn.
(cid:3)
Example 2.8 (continued). For the points in Example 2.3 we set
h = x + z−3x + z−6x ,
1 1 2 3
h = x + z−5x + z−2x ,
2 1 2 3
h = x + x + z−1x , and
3 1 2 3
h = z−1x + z−5x + x .
4 1 2 3
TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 7
Then h = h ·h ·h ·h equals
1 2 3 4
h = z−1x 4+(z−6 +z−5+z−4+z−1)x 3x +(z−7 +z−3+z−2+1)x 3x +
1 1 2 1 3
(z−10+z−9+z−8+z−6+z−5+z−4)x 2x 2+(z−12+z−11+3z−7+2z−6+
1 2
2z−5+2z−3+1)x 2x x +(z−9+z−8+z−6+z−4+z−2+z−1)x 2x 2+(z−13+
1 2 3 1 3
z−10+z−9+z−8)x x 3+(z−16+z−12+2z−11+2z−10+z−9+z−8+z−7+
1 2
z−6+z−5+z−3)x x 2x +(2z−13+z−12+z−11+z−9+z−8+z−7+2z−6+
1 2 3
z−5+z−4+z−2)x x x 2+(z−10+z−8+z−7+z−3)x x 3+z−13x 4+(z−16+
1 2 3 1 3 2
z−14+z−10+z−8)x 3x +(z−17+z−13+2z−11+z−9+z−5)x 2x 2+(z−14+
2 3 2 3
z−12+z−8+z−6)x x 3+z−9x 4.
2 3 3
Its max-tropicalization is
tropmax(h) = (−1⊙x ⊙4)⊕(−1⊙x ⊙3⊙x )⊕(x ⊙3⊙x )⊕(−4⊙x ⊙2⊙
1 1 2 1 3 1
x ⊙2)⊕(x ⊙2⊙x ⊙x )⊕(−1⊙x ⊙2x ⊙2)⊕(−8⊙x ⊙x ⊙3)⊕(−3⊙x ⊙
2 1 2 3 1 3 1 2 1
x ⊙2⊙x )⊕(−2⊙x ⊙x ⊙x ⊙2)⊕(−3⊙x ⊙x ⊙3)⊕(−13⊙x ⊙4)⊕(−8⊙
2 3 1 2 3 1 3 2
x ⊙3⊙x )⊕(−5⊙x ⊙2⊙x ⊙2)⊕(−6⊙x ⊙x ⊙3)⊕(−9⊙x ⊙4).
2 3 2 3 2 3 3
The polynomial h and its max-tropicalization have 15 terms each, one for each maximal cell
of thearrangement shownin Figure1. For instance, thepointp = (0,1,0) is generic with fine
type T(p) = (12,3,4). Evaluating tropmax(h) at p gives −1, and this maximum is attained
for the unique tropical monomial x ⊙2 ⊙x ⊙x = 2x +x +x . We have t(p) = (2,1,1)
1 2 3 1 2 3
for the coarse type.
Recall that acomposition of n into dparts is asequence (t ,t ,...,t )∈ Nd of nonnegative
1 2 d
integerssuchthatt +t +···+t = n. Suchcompositionsbijectivelycorrespondtomonomials
1 2 d
of total degree n in d variables, of which there are exactly n+d−1 = n+d−1 .
n d−1
(cid:0) (cid:1) (cid:0) (cid:1)
Theorem 2.9. For an arrangement A = A(V), the map t from the set of cells in C of
A
maximal dimension d−1 to the set of compositions of n into d parts is injective. Moreover,
if the points V are sufficiently generic then the map t is bijective.
Proof. The injectivity of t follows immediately from Proposition 2.7.
Now suppose that the points in V are sufficiently generic. All coordinates are finite, and
hence all linear monomials are present in the linear form h with non-zero coefficients. Since
i
h is the product of h ,h ,...,h all possible monomials of degree n in the d indeterminates
1 2 n
x ,x ,...,x actuallyoccurinh. Thereforealltropicalmonomialsoccurinthetropicalization
1 2 d
tropmax(h). As we assumed that the coefficients in V are chosen sufficiently generic, the
coefficients in the tropical polynomial tropmax(h) are generic. Equivalently, each monomial
of h defines a facet of P , and hence the claim. (cid:3)
A
Proposition4.2providesaninterpretationoftheaboveresultintermsofmixedsubdivisions
of dilated simplices.
Corollary 2.10. For an arrangement A = A(V), the number of cells in C of maximal
A
dimension d−1 does not exceed n+d−1 . If the points V are sufficiently generic then this
n
bound is attained. (cid:0) (cid:1)
Note that the injectivity of t does not extend to the lower-dimensional cells, even in the
sufficiently generic case. For instance, in thearrangementconsidered inExamples 2.3 and2.8
thepoints(0,2,3) and(0,3,5) lieintherelativeinteriorsoftwodistinct1-cells; yettheyshare
thesamecoarse type(1,1,3). However coarse types arelocally distinctinthefollowing sense.
Proposition 2.11. Let C and D be two distinct cells in C . If C is contained in the closure
A
of D then C and D have distinct coarse types.
8 ANTONDOCHTERMANN,MICHAELJOSWIG, ANDRAMANSANYAL
Proof. By [7, Cor. 13] we have T (D) ≤ T (C) and T (D) 6= T (C). This, in particular
A A A A
implies that C is contained in more closed sectors with respect to A and hence C and D
cannot have the same coarse type. (cid:3)
Note also that the boundedness of a cell in C can be read off its coarse type.
A
Proposition 2.12 ([7, Cor. 12]). Let A = A(V) be a tropical hyperplane arrangement and
let C ∈ C be a cell in the induced decomposition. Then C is bounded if and only if t(C) > 0
A i
for all i = 1,...,n.
Proof. We noted previously that for two points p,q ∈ Td−1 we have p ∈ Smax(q) if and only
k
if q ∈ Smin(p). Now, the point p is contained in an unbounded cell of C if there is a k such
k A
thatSmin(p)∩V = ∅. Thisisthecase ifandonly if t (p) = 0, thatis, thereisnohyperplane
k A k
H for which p is contained in the k-th sector. (cid:3)
i
Remark 2.13. There is another perspective on Proposition 2.7 that we wish to share. The
standard valuation ν : (C{{z}}gen)∗ → R on the field of (generalized) Puiseux series maps
an element to its order. For ω ∈ Rd, the ω-weight of f = c xa ∈ C{{z}}gen[x ,...,x ]
a a 1 d
is ω(f) = max{ν(ca)+hω,ai : ca 6= 0} and the initial forPm of the polynomial f is given
by in (f) := {xa : ν(c )+hω,ai = ω(f)}. Finally, for an ideal I ⊂ C{{z}}gen[x ,...,x ]
ω a 1 d
the initial idePal inω(I) is generated by all inω(f) for f ∈ I. An alternative characterization
of the tropical variety was given by Speyer and Sturmfels [30, Thm. 2.1] as the collection of
weights ω ∈ Rd such that in (I) does not contain a monomial. For a tropical hyperplane
ω
arrangement A = A(V) with defining polynomial h = h ∈ C{{z}}gen[x ,...,x ] the coarse
A 1 d
type t(p) of a generic point p ∈ Td−1\A satisfies
hxt(p)i = in (hhi).
p
Remark 2.14. The evaluation of the tropical polynomial tropmax(h) at some point p can
be modeled as a min-cost-flow problem as follows: Define a directed graph with nodes α,
β ,β ,...,β , γ ,γ ,...,γ , and δ. For each i ∈ [n] and each k ∈ [d] there are the following
1 2 n 1 2 d
arcs: from α to β with cost 0 and capacity 1, from β to γ with cost v and capacity 1,
i i k ik
from γ to δ with cost p and unbounded capacity. Now tropmax(h)(p) equals the minimal
k k
cost of shipping n units of flow from the unique source α to the unique sink δ. This problem
has a strongly polynomial time solution in the parameters n and d; see [28, Chapter 12].
This is remarkable in view of the fact that the tropical polynomial tropmax(h) may have
exponentially many monomials.
2.3. Topology of types. We next investigate topological properties of certain subsets of
Td−1 induced by a tropical hyperplane arrangement. The motivation for such a study comes
from our applications to (co)cellular resolutions as describedin Section 3. However, since the
methods used to establish the desired properties are based on notions from (coarse) tropical
convexity, we include them here. In particular, we show that certain subsets of Td−1 are
contractible by proving that they are, in fact, tropically convex. Let us emphasize that none
of the results that follow require the hyperplanes to be in general position.
We begin with the following observation, which was established in [7, Thm. 2]. We include
a short proof for the sake of completeness.
Proposition 2.15. A tropically convex set is contractible.
Proof. The distance of two points p,q ∈ Td−1 is defined as
dist(p,q) := max |p −p +q −q |.
i j j i
1≤i<j≤d
TROPICAL TYPES AND ASSOCIATED CELLULAR RESOLUTIONS 9
We have the equations
dist(p,q)⊙p ⊕q = q and p⊕ dist(p,q)⊙q = p.
(cid:0) (cid:1) (cid:0) (cid:1)
Now let p be a point in some tropically convex set S. The map
η : S ×[0,1] → S : (q,t) 7→ ((1−t)·dist(p,q))⊙p ⊕ (t·dist(p,q))⊙q
(cid:0) (cid:1) (cid:0) (cid:1)
is continuous, and it contracts the set S to the point p. (cid:3)
Recall that for two types T,T′ ∈ {0,1}n×d we writeT ≤ T′ for the componentwise induced
partial order. We let min(T,T′) and max(T,T′) denote the tables with entries given by the
componentwise minimum and maximum, respectively.
Proposition 2.16. Let A = A(V) be a max-tropical hyperplane arrangement in Td−1 and
p,q ∈ Td−1. Then
min(T (p),T (q)) ≤ T (r) ≤ max(T (p),T (q))
A A A A A
for every point r ∈ tconvmax{p,q} on the max-tropical line segment between p and q.
Proof. Let r = (λ ⊙ p)⊕ (µ ⊙ q) = max{λ + p,µ + q} for λ,µ ∈ R and let k ∈ [d] be
1 1
arbitrary but fixed. We treat each inequality separately but in each case we assume without
loss of generality that r = λ+p ≥ µ+q .
k k k
For thefirstinequality supposethatbothp andq arein thek-th sector of somehyperplane
H(v), so that p −p ≥ v −v and q −q ≥ v −v for all i ∈ [d]. Now, for j ∈ [d] we
k i k i k i k i
distinguish two cases. If r = λ+p ≥ µ+q , then r −r = p −p , so that r is in the k-th
j j j j k j k
sector of H(v). If r = µ+q ≥ λ+p , then r −r ≥ µ+q −r =µ+q −(µ+q ) = q −q .
j j j k j k j k j k j
Hence r −r ≥ v −v for all j ∈ [d], and we conclude that r is in the k-th sector of H(v).
k j k j
For thesecond inequality supposethat r is contained in thek-th sector of somehyperplane
H(v), so that r −r ≥ v −v for all j ∈ [d]. Since r = λ +p ≥ µ+q we have that
k j k j k k k
λ+p ≥ v −v +r for all j ∈ [d]. Also, r ≥λ+p and hence λ+p ≥ v −v +λ+p . We
k k j j j j k k j j
conclude p −p ≥ v −v for all j ∈[d]. Hence p is in the k-th sector of H(v), as desired. (cid:3)
k j k j
From the definition the coarse type we obtain the following statement regarding coarse
types.
Corollary 2.17. Let A = A(V) be a max-tropical hyperplane arrangement in Td−1 and
p,q ∈ Td−1. Then
t (r) ≤ max(t (p),t (q))
A A A
for r ∈ tconvmax{p,q}.
From Proposition 2.16 and Corollary 2.17 we obtain the following result regarding the
topology of regions of bounded fine and coarse (co)type. These will be the main tools for
establishing results regarding (co)cellular resolutions in Section 3.
Corollary 2.18. Let A = A(V) be an arrangement of n max-tropical hyperplanes in Td−1,
and let B ∈ {0,1}n×d and b ∈ Nd. With labels determined by fine (respectively, coarse) type
the following subsets of Td−1 are max-tropically convex and hence contractible:
(C ,T) := p ∈ Td−1 : T (p)≤ B = {C ∈ C : T (C)≤ B}
A ≤B A A A
(CA,t)≤b := (cid:8)p ∈ Td−1 : tA(p) ≤ b (cid:9) = S{C ∈ CA : tA(C) ≤ b} .
(cid:8) (cid:9) S
10 ANTONDOCHTERMANN,MICHAELJOSWIG, ANDRAMANSANYAL
Similarly, with labels determined by fine (respectively, coarse) cotype the following subsets of
Td−1 are min-tropically convex and hence contractible:
(C ,T) := p ∈ Td−1 : T (p)≤ B = C ∈ C : T (C)≤ B
A ≤B A A A
(CA,t)≤b := (cid:8)p ∈ Td−1 : tA(p) ≤ b (cid:9) = S(cid:8)C ∈ CA : tA(C)≤ b (cid:9).
(cid:8) (cid:9) S(cid:8) (cid:9)
As a consequence, the two subsets of Td−1 obtained by replacing the complex C in the
A
above pair of formulas with the bounded complex B are min-tropically convex and hence
A
contractible.
Proof. The max-tropical convexity of (C ,T) follows from Proposition 2.16, and Corol-
A ≤B
lary 2.17 establishes the same property for the coarse variant (CA,t)≤b.
If r is a point in the min-tropical line segment between p and q a reasoning similar to
the proof of Proposition 2.16 shows that T (r) ≥ min(T (p),T (q)). Passing to comple-
A A A
ments yields T (r) ≤ max(T (p),T (q)) and thus t (r)≤ max(t (p),t (q)) for the coarse
A A A A A A
types. We conclude that the sets (CA,T)≤B and (CA,t)≤b are min-tropically convex. For
the last claim, we note that the tropical complex B itself is min-tropically convex; in view
A
of Proposition 2.12 it is a down-set of C under the cotype labeling. Hence both (B ,T)
A A ≤B
and (BA,t)≤b are intersections of min-tropically convex sets. (cid:3)
Note that the sets of bounded (coarse) type need not be closed or bounded. In Figure 2
we illustrate an example of a coarse down-set from our running example.
7
004 013
v1
6
5
4
103 022
3
112
v2
2
202
v3 031
1
301 211 121
0
-1
v4
400 310 220 130 040
-2
-1 0 1 2 3 4 5 6
Figure 2. Coarse down-set (C ) for the arrangement A from Example 2.3.
A ≤(2,2,2)
3. Resolutions
In this section we show how the polyhedral complexes C and B arising from a tropical
A A
hyperplane arrangement A = A(V) support resolutions for associated monomial ideals. We
begin with a few definitions.