Coamoebas of polynomials supported on circuits

0.65 MB·

by  Jens Forsgård

COAMOEBAS OF POLYNOMIALS SUPPORTED ON CIRCUITS
JENSFORSG˚ARD
6
1 Abstract. We study coamoebas of polynomials supported on circuits. Our
0 resultsincludeanexplicitdescriptionofthespaceofcoamoebas,arelationbe-
2 tweenconnectedcomponentsofthecoamoebacomplementandcriticalpoints
of the polynomial, an upper bound on the area of a planar coamoeba, and a
n
recoveredboundonthenumberofpositivesolutionsofafewnomialsystem.
a
J

0
2
1. Introduction
]
G ApossiblydegeneratecircuitisapointconfigurationA⊂Zn ofcardinalityn+2
A which span a sublattice ZA of rank n. That is, such that the Newton polytope
h. NA = Conv(A) is of full dimension. A polynomial system f(z) = 0 is said to be
supported on a circuit A if each polynomial occurring in f(z) is supported on A.
t
a Polynomial systems supported on circuits have recently been been studied in the
m
context of, e.g., real algebraic geometry [4, 6], complexity theory [5], and amoeba
[ theory[18]. Thename“circuit”originatefrommatroidtheory; see[17]and[20]for

further background.
1
v The aim of this article is to describe geometrical and topological properties of
8 coamoebas of polynomials supported on circuits. Such an investigation is moti-
6 vated not only by the vast number of applications of circuits in different areas of
4
geometry, but also since circuits provide an ideal testing ground for open problems
5
in coamoeba theory.
0
. This paper is organized as follows. In Section 2 we will give a brief overview of
1
coamoeba theory. In Section 3 we will discuss the relation between real polynomi-
0
6 als and the coamoeba of the A-discriminant. The main results of this paper are
1 contained in Sections 4–7, each of which can be read as a standalone text.
: InSection4wewillgiveacompletedescriptionofthespaceofcoamoebas. That
v
i is,wewilldescribehowthetopologyofthecoamoebaCf dependsonthecoefficients
X
of f. Describing the space of amoebas is the topic of the articles [18] and [19], and
r to fully appreciate our result one should consider these spaces simultaneously, see
a
e.g. Figure 2. The geometry of the space of coamoebas is closely related to the
A-discriminantal variety, see Theorems 4.1 and 4.2.
InSection5wewillprovethattheareaofaplanarcircuitcoamoebaisbounded
from above by 2π2. That is, a planar circuit coamoeba covers at most half of
the torus T2. Furthermore, we will prove that a circuit admits a coamoeba of
maximal area if and only if it admits an equimodular triangulation. Note that
we calculate area without multiplicities, in contrast to [11]. However, the relation
between (co)amoebas of maximal area and Harnack curves is made visible also in
this setting.
Date:January22,2016.
1
2 JENSFORSG˚ARD
In Section 6 we will prove that, under certain assumptions on A, the critical
pointsoff(z)areprojectedbythecomponentwiseargumentmappingintodistinct
connected component of the complement of the coamoeba C . Furthermore, this
f
projection gives a bijective relation between the set of critical points and the set
of connected components of the complement of the closed coamoeba. This settles
a conjecture used in in [10] when computing monodromy in the context of dimer
models and mirror symmetry.
In Section 7 we will consider bivariate systems supported on a circuit. If such a
system is real, then it admits at most three roots in R2. The main contribution of
+
thissectionisthatweofferanewapproachtofewnomialtheory. Usingourmethod,
we will prove that if N is a simplex, then, for each θ ∈ T2, a complex bivariate
A
system supported on A has at most two roots in the sector Arg−1(θ).
2. Coamoebas and lopsidedness
Let A denote a point configuration A = {a ,...,a } ⊂ Zn, where N = #A.
0 N−1
By abuse of notation, we identify A with the (1+n)×N-matrix
(cid:18) (cid:19)
1 ... 1
(1) A= .
a ... a
0 N−1
Thecodimension ofAistheintegerm=N−1−n. Acircuit isapointconfiguration
of codimension one. A circuit is said to be nondegenerate if it is not a pyramid
over a circuit of smaller dimension. That is, if all maximal minors of the matrix
A are nonvanishing. We will partition the set of circuits into two classes; simplex
circuits, for which N is a simplex, and vertex circuits, for which A=Vert(N ).
A A
We associate to A the family CA consisting of all polynomials
∗
N−1
(cid:88)
f(z)= fkzak,
k=0
where f(z) is identified with the point f = (f ,...,f ) ∈ CA. By slight abuse
0 N−1 ∗
of notation, we will denote by fk(z) the monomial function z (cid:55)→fkzak. We denote
the algebraic set defined by f by Z(f) ⊂ Cn. The coamoeba C is the image of
∗ f
Z(f) under the componentwise argument mapping Arg: Cn →Tn defined by
∗
Arg(z)=(arg(z ),...,arg(z )),
1 n
whereTn denotestherealn-torus. Itissometimesbeneficialtoconsiderthemulti-
valued argument mapping, which gives the coamoeba as a multiply periodic subset
of Rn. Coameobas were introduced by Passare and Tsikh as a dual object, in an
imprecise sense, of the amoeba A .
f
We will say that a point z ∈Cn is a critical point of f if it solves the system
∗
(2) ∂ f(z)=···=∂ f(z)=0.
1 n
If in addition z ∈ Z(f) then z will be called a singular point of f. The A-
discriminant ∆(f) = ∆ (f) is an irreducible polynomial with domain CA which
A ∗
vanishes if and only if f has a singular point in Cn [9].
∗
A Gale dual of A is an integer matrix B whose columns span the right Z-kernel
of A. That is, B is an integer N ×m-matrix, of full rank, such that its maximal
minors are relatively prime. A Gale dual is unique up to the action of SL (Z).
m
COAMOEBAS OF POLYNOMIALS SUPPORTED ON CIRCUITS 3
The rows b of B are indexed by the points a ∈A. To each Gale dual we associate
k
a zonotope
(cid:40) N−1 (cid:12) (cid:41)
ZB = π2 (cid:88) λkbk(cid:12)(cid:12)(cid:12)|λk|≤1 ⊂Rm.
k=0
We will say that a triangulation T of N is a triangulation of A if Vert(T)⊂A.
A
Such a triangulation is said to be equimodular if all maximal simplices has equal
volume.
Lethbeaheight function h: A→R. ThefunctionhinducesatriangulationT
h
of A in the following manner. Let N denote the polytope in Rn+1 with vertices
h
(a,h(a)). The lower facets of N are the facets whose outward normal vector
h
has negative last coordinate. Then, T is the triangulation of A whose maximal
h
simplices are the images of the lower facets of N under the projection onto the
h
first n coordinates. A triangulation T of A is said to be coherent if there exists a
height function h such that T =T .
h
If A is a circuit then B is a column vector, unique up to sign. Hence, the
zonotope Z is an interval. Let A =A\{a }, with associated matrix A , and let
B k k k
V = Vol(A ). If A is a nondegenerate circuit, so that V > 0 for all k, then N
k k k A
admitsexactlytwocoherenttriangulationswithverticesinA[9]. Denotethesetwo
triangulations by T for δ ∈{±1}. Each simplex N occurs in exactly one of the
δ Ak
triangulations T . Hence, there is a well-defined assignment of signs k (cid:55)→δ , where
δ k
δ ∈{±1}, such that
k
(cid:8) (cid:9)
T = N , δ =±1.
δ Ak δk=δ
Here,wehaveidentifiedatriangulationwithitssetofmaximalsimplices. Asshown
in [9, chp. 7 and chp. 9] and [7, sec. 5], a Gale dual of A is given by
(3) b =(−1)k|A |=δ V .
k k k k
Thus, the zonotope Z is an interval of length 2πVol(A).
B
TheA-discriminant∆hasn+1homogeneities,oneforeachrowofthematrixA.
EachGaledualcorrespondtoadehomogenizationof∆. Tobespecific,introducing
the variables
N−1
(4) ξ = (cid:89) fbkj, j =1,...,m,
j k
k=0
there is a Laurent monomial M(c) and a polynomial ∆ (ξ) such that
B
∆ (ξ)=M(f)∆ (f).
B A
Wewillsaythat∆ isthereduced form of∆. Suchareductionyieldsaprojection
B
pr : CA → Cm, and we will say that Cm is the reduced family associated to A,
B ∗ ∗ ∗
and that pr (f) is the reduced form of f.
B
Example 2.1. Let A = {0,1,2}, so that CA is the family of quadratic univariate
∗
polynomials
f(z)=f +f z+f z2.
0 1 2
ConsidertheGaledualB =(1,−2,1)t,andintroducethevariableξ =f f−2f . In
0 1 2
this case the A-discriminant ∆ is well-known, and we find that
A
f−2∆ (f)=f−2(cid:0)f2−4f f (cid:1)=1−4ξ =∆ (ξ).
1 A 1 1 0 2 B
4 JENSFORSG˚ARD
The projection pr correspond to performing the change of variables z (cid:55)→f f−1z,
B 0 1
and multiplying f(z) by f−1, after with we obtain the reduced family consisting of
0
all polynomials of the form
f(z)=1+z+ξz2.
LetS denoteasubsetofA. Thetruncated polynomial f istheimageoff under
S
the projection pr : CA → CS. Of particular interest is the case when S = Γ∩A
S ∗ ∗
for some face Γ of the Newton polytope N (denoted by Γ ≺ N ). We will write
A A
f =f . It was shown in [14] that
Γ Γ∩A
(cid:91)
C = C ,
f fΓ
Γ≺NA
Let E denote the set of edges of N , then the shell of the coamoeba is defined by
A
(cid:91)
H = C .
f fΓ
Γ∈E
As an edge Γ is one dimensional, the shell H is a hyperplane arrangement. Its
f
importancecanbeseeninthateachfull-dimensionalcellofH containatmostone
f
connected component of the complement of C , see [7].
f
Example 2.2. The coamoeba of f(z)=1+z +z , as described in [7] and [14], can
1 2
be seen in Figure 1, where it is drawn in the fundamental domains [−π,π]2 and
[0,2π]2. TheshellH consistofthehyperplanearrangementdrawninblack. Inthis
f
case, it is equal to the boundary of C . The Newton polytope N and its outward
f A
normal vectors are drawn in the rightmost picture. If H is given orientations
f
in accordance with the outward normal vectors of N , then the interior of the
A
coamoeba consist of the oriented cells.
Figure 1. Thecoamoebaoff(z)=1+z +z intwofundamental
1 2
regions, and the Newton polytope N .
A
Acting on A by an integer affine transformation is equivalent to performing a
monomial change of variables and multiplying f by a Laurent monomial. Such an
action induces a linear transformation of the coamoeba C , when viewed in Rn [7].
f
We will repeatedly use this fact to impose assumptions on A, e.g., that it contains
the origin.
Thepolynomialf issaidtobecolopsided atapointθ ∈Rn ifthereexistaphase
ϕ such that
(5) (cid:60)(cid:0)eiϕf (eiθ)(cid:1)≥0, k =0,...,N −1,
k
COAMOEBAS OF POLYNOMIALS SUPPORTED ON CIRCUITS 5
with at least one of the inequalities (5) being strict. The motivation for this defi-
nition is as follows. If f is colopsided at θ, then
N−1
(cid:60)(cid:0)eiϕf(reiθ)(cid:1)= (cid:88) rak(cid:60)(cid:0)eiϕfk(eiθ)(cid:1)>0, ∀r ∈Rn+,
k=0
since at least one term of the sum is strictly positive. Hence, colopsidedness at θ
implies that θ ∈Tn\C . The colopsided coamoeba, denoted L , is defined as the
f f
set of all θ such that (5) does not hold for any phase ϕ [7]. Hence, C ⊂L .
f f
Each monomial f (z) defines an affinity (i.e., a group homomorphism composed
k
with a translation) fk: Cn∗ → C∗ by z (cid:55)→ fkzak. We thus obtain unique affinities
|f | and fˆ such that the following diagram of short exact sequences commutes:
k k
0 Rn Cn (S1)n 0
+ ∗
|fk| fk fˆk
0 R+ C∗ S1 0.
Notice that T (cid:39) S1 ⊂ C. We denote by fˆ(θ) ⊂ (S1)A ⊂ CA the vector with
∗
components fˆ(θ). Assume that f contains the constant monomial 1, and consider
k
the map ord (f): Rn →Rm defined by
B
(6) ord (f)(θ)=Arg (cid:0)fˆ(θ)(cid:1)·B,
B π
where Arg denotes the componentwise principal argument map. It was shown in
π
[7] that the map ord (f) induces a map
B
(7) ord (f): Tn\L →{Arg (f)B+2πZm}∩intZ .
B f π B
which in turn induces a bijection between the set of connected components of the
complementofL andthefinitesetintherighthandsideof (7). Themapord (f)
f B
is known as the order map of the lopsided coamoeba.
Remark 2.3. The requirement that f contains the monomial 1 is related to the
choice of branch cut of the function Arg; in order to obtain a well-defined map,
we need the right hand side of (6) to be discontinuous only for θ such that two
components of fˆ(θ) are antipodal, see [7]. If f does not contain the constant
monomial 1, then one should fix a point a ∈ A and multiply the vector fˆ(θ) by
k
the scalar fˆ(θ)−1 before taking principal arguments. It is shown in [7, thm. 4.3]
k
that the obtained map is independent of the choice of a .
k
If θ ∈Tn\L , then we can choose ϕ such that
f
(cid:60)(cid:0)eiϕf (eiθ)(cid:1)>0, k =0,...,N −1.
k
That is, the boundary of L is contained in the hyperplane arrangement consisting
f
of all θ such that two components of fˆ(θ) are antipodal.
Ithasbeenconjecturedthatthenumberofconnectedcomponentsofthecomple-
mentofC isatmostVol(A).1 Aproofinarbitrarydimensionhasbeenproposedby
f
Nissein[13],andanindependentproofinthecasen=2wasgivenin[8]. Thatthe
1ThisconjecturehascommonlybeenattributedtoMikaelPassare,however,itseemstoorigi-
natefromatalkgivenbyMounirNisseatStockholmUniversityin2007.
6 JENSFORSG˚ARD
number of connected components of the complement of L is at most Vol(A) fol-
f
lowsfromthetheoryofMellin–BarnesintegralrepresentationsofA-hypergeometric
functions, see [2] and [3].
A finite set I ⊂ Tn which is in a bijective correspondence with the set of con-
nected components of the complement of C by inclusion, will be said to be an
f
index set of the coamoeba complement. This notation will be slightly abused; a
set I of cardinality Vol(A) will be said to be an index set of the coamoeba if each
connected component of its complement contains exactly one element of I.
Theterm“lopsided”wasfirstusedbyPurbhooin[15],denotingthecorrespond-
ingconditionto(5)foramoebas: thepolynomialf issaidtobelopsided atapoint
x∈Rn if there is a a ∈A such that the moduli |f |(x) is greater than the sum of
k k
the remaining modulis. As a comparison, note that the polynomial f is colopsided
at θ ∈T if and only if the greatest intermediate angle of the components of fˆ(θ) is
greater than the sum of the remaining intermediate angles.
3. Real points and the coameoba of the A-discriminant
We will say that f is real at θ, if there is a real subvector space (cid:96)⊂C such that
fˆ(θ) ∈ (cid:96) for all k = 0,...,N −1. If such a θ exist then f is real, that is, after a
k
change of variables and multiplication with a Laurent monomial f ∈ RA. In this
∗
section, we will study the function fˆfrom the viewpoint of real polynomials. Our
mainresultisthefollowingcharacterizationofthecoamoebaoftheA-discriminant
of a circuit.
Proposition 3.1. Let A be a nondegenerate circuit, and let δ be as in (3). Then,
k
Arg(f) ∈ C if and only if after possibly multiplying f with a constant, there is a
∆
θ ∈Rn such that fˆ(θ)=δ for all k.
k k
IfAisacircuitandB isaGaledualofAthentheHorn–Kapranovparametriza-
tion of the reduced discriminant ∆ can be lifted to a parametrization of the
B
discriminant surface ∆ as
z (cid:55)→(b za0,...,b zaN−1).
0 N−1
Taking componentwise arguments, we obtain a simple proof Proposition 3.1. In
particular, the proposition can be interpreted as a coamoeba version of the Horn–
Kapranov parametrization valid for circuits. Our proof of Proposition 3.1 will be
more involved, however, for our purposes the lemmas contained in this section are
of equal importance.
Lemma 3.2. Assume that the polynomial f is real at θ ∈Rn. Then, f is real at
0
θ ∈Rn if and only if θ ∈θ +πL, where L is the dual lattice of ZA.
0
Proof. After translating θ and multiplying f with a Laurent monomial, we can
assume that θ =0, that (cid:96) =R, and that f contains the monomial 1. That is, all
0 0
coefficients of f are real, in particular proving if-part of the statement. To show
the only if-part, notice first that fˆ(θ) ⊂ (cid:96) implies that (cid:96) contains both the origin
and 1. That is, (cid:96) = R. Furthermore, fˆ(θ) ⊂ R only if for each a ∈ A there is a
k ∈Z such that (cid:104)a,θ(cid:105)=πk, which concludes the proof. (cid:3)
COAMOEBAS OF POLYNOMIALS SUPPORTED ON CIRCUITS 7
The A-discriminant ∆ related to a circuit has been described in [9, chp. 9, pro.
1.8] where the formula
(cid:89) (cid:89) (cid:89) (cid:89)
(8) ∆(f)= bbk f−bk − b−bk fbk
k k k k
δk=1 δk=−1 δk=−1 δk=1
was obtained. In particular, ∆ is a binomial. As the zonotope Z is a symmetric
B
interval of length 2πVol(A), the image of the map ord (f) is of cardinality Vol(A)
B
unless
(9) Arg (f)B ≡2πVol(A) mod 2π.
π
In particular, the complement of C has the maximal number of connected compo-
f
nents (i.e., Vol(A)-many) unless the equivalence (9) holds.
Lemma 3.3. For each κ = 0,1,...,n+1, there are exactly Vol(A )-many points
κ
θ ∈T such that
(10) fˆ(θ)=δ , ∀k (cid:54)=κ.
k k
Proof. Byapplyinganintegeraffinetransformation,thestatementfollowsfromthe
case when A consist of the vertices of the standard simplex. (cid:3)
κ
Lemma 3.4. Fix κ ∈ {0,...,n+1}. For each θ fulfilling (10), let ϕ ∈ T be
θ
defined by the condition that if arg (f )=ϕ then
π κ θ
(11) fˆ(θ)=δ .
κ κ
Assume that ZA=Zn. Then, the numbers ϕ are distinct.
θ
Proof. Wecanassumethata =0andthatf =1. Assumethatϕ =ϕ . Then,
0 0 θ1 θ2
(cid:104)a,θ (cid:105)=(cid:104)a,θ (cid:105)+2πr, ∀a∈A.
2 1
By translating, we can assume that θ =0, and hence, since 1 is a monomial of f,
1
that all coefficients are real. Consider the lattice L consisting of all points θ ∈ Rn
suchthatf isrealatθ. SinceZA=Zn,Lemma3.2showsthatL=πZn. However,
we find that
(cid:68) θ (cid:69)
a, 2 =πr,
2
and hence θ22 ∈L. This implies that θ2 ∈2πZn, and hence θ2 =0 in Tn. (cid:3)
Proof of Proposition 3.1. Assume first that there is a θ as in the statement of the
proposition, where we can assume that θ = 0. Then, arg(f ) = arg(δ ). It follows
k k
that the monomials
(cid:89) (cid:89) (cid:89) (cid:89)
bbk f−bk and bbk f−bk
k k k k
δk=1 δk=−1 δk=−1 δk=1
haveequalsigns. Therefor,∆vanishesforf =δ |b |,implyingthatArg(f)∈C .
k k k ∆
For the converse, fix κ, and reduce f by requiring that f = δ |b | = b for
k k k k
k (cid:54)= κ. Let I denote the set of points θ ∈ Tn such that fˆ(θ) = δ for k (cid:54)= κ,
k k
which by Lemma 3.3 has cardinality V . By Lemma 3.4, the set I is in a bijective
κ
correspondence with values of arg(f ) such that fˆ(θ)=δ . Therefor, we find that
κ κ κ
∆ vanishes at f =V eiϕ for each ϕ∈I. However, the discriminant ∆ specializes,
κ κ
up to a constant, to the binomial
∆ (f )=f|bκ|−b|bκ| =fVκ −bVκ,
κ κ κ κ κ κ
8 JENSFORSG˚ARD
which has exactly V -many solutions in C of distinct arguments. Hence, since
κ ∗
∆(f) = 0 by assumption, and comparing the number of solutions, it holds that
fˆ(θ)=δ for one of the points θ ∈I. (cid:3)
κ κ
4. The space of coamoebas
Let U ⊂CA denote the set of all f such that the number of connected compo-
k ∗
nents of the complement of C is Vol(A)−k. Describing the sets U is known as
f k
the problem of describing the space of coamoebas of CA. In this section, we will
∗
give explicit descriptions of the sets U in the case when A is a circuit. As a first
k
observation we note that the image of the map ord (f) is at least of cardinality
B
Vol(A)−1, implying that
CA =U ∪U ,
∗ 0 1
and in particular U = ∅ for k ≥ 2. Hence, it suffices for us to give an explicit
k
descriptionofthesetU . Ourmainresultisthefollowingtwotheorems,highlighting
1
also the difference between vertex circuits and simplex circuits. Note that ∆ is a
real polynomial [9].
Theorem 4.1. Assume that A is a nondegenerate simplex circuit, with a as
n+1
an interior point. Choose B such that δ = −1, and let ∆ be as in (8). Then,
n+1
f ∈U if and only if Arg(f)∈C and
1 ∆
(12) (−1)Vol(A)∆(cid:0)δ |f |,...,δ |f |(cid:1)≤0.
0 0 n+1 n+1
Theorem 4.2. Assume that A is a vertex circuit. Then, f ∈ U if and only if
1
Arg(f)∈C .
∆
The article [18] describes the space of amoebas in the case when A is a simplex
circuitindimensionatleasttwo. Inthiscase,thenumberofconnectedcomponents
of the amoeba complement is either equal to the number of vertices of N or one
A
greater. One implication of [18, thm. 4.4 and thm. 5.4] is that, if the amoeba
complement has the minimal number of connected components, then
(−1)Vol(A)∆(cid:0)δ |f |,...,δ |f |(cid:1)≥0.
0 0 n+1 n+1
Furthermore, this set intersect U only in the discriminant locus ∆(f) = 0. The
1
space of amoebas in the case when A is a simplex circuit in dimension n = 1 has
been studied in [19], and is a more delicate problem. On the other hand, if A
is a vertex circuit, then each f ∈ CA is maximally sparse and hence has a solid
∗
amoeba. Thatis,thecomponentsofthecomplementoftheamoebaisinabijective
correspondance with the vertices of the Newton polytope N . In particular, the
A
number of connected components of the amoeba complement does not depend on
f. From Theorems 4.1 and 4.2 we see that a similar discrepancy between simplex
circuits and vertex circuits occur for coamoebas.
Example 4.3. The reduced family
f(z)=1+z3+z3+ξz z .
1 2 1 2
was considered in [16, ex. 6, p. 59], where the study of the space of amoebas was
initiated. We have drawn the space of amoebas and coamoebas jointly in the left
picture in Figure 2. The blue region, whose boundary is a hypocycloid, marks
values of ξ for which the amoeba complement has no bounded component. The set
U is seen in orange. The red dots is the discriminant locus ∆(ξ) = 0, which is
1
contained in the circle |ξ|=3 corresponding an equality in (12).
COAMOEBAS OF POLYNOMIALS SUPPORTED ON CIRCUITS 9
Example 4.4. The reduced family
f(z)=1+z +z3+ξz3z
1 2 1 2
is a vertex circuit. In this case, the topology of the amoeba does not depend on
the coefficient ξ. The space of coamoebas is drawn in the right picture in Figure 2.
ThesetU comprisesthethreeorangelinesemergingfromtheorigin. Thereddots
1
is the discriminant locus ∆(ξ) = 0. It might seem like the set U is disconnected,
0
however this a consequence of that we consider f in reduced form. In CA the set
∗
U is connected, though not simply connected.
0
Figure 2. The amoeba and coamoeba spaces of Examples 4.3 and 4.4.
4.1. Proof of Theorem 4.1. Impose the assumptions of Theorem 4.1. Then,
b +···+b =−b =Vol(A),
0 n n+1
and in particular V = Vol(A). By Lemma 3.3 there is a set I of cardinality
n+1
Vol(A) consisting of all points θ such that fˆ(θ) = ··· = fˆ (θ) = δ = 1 In
0 n k
particular, f is colopsided at θ ∈ I unless fˆ (θ) = −1. It was shown in [7, sec.
n+1
5] that, if f (cid:54)∈C , then I is an index set for the complement of C . In fact, I is an
∆ f
index set of the complement of C for arbitrary f.
f
Proposition 4.5. Let A be a simplex circuit. Assume that Arg(f) ∈ C , i.e.,
∆
that there exists a θ ∈ I with fˆ (θ) = δ . Then, the complement of C has
n+1 n+1 f
Vol(A)-many connected components if and only if it contains θ.
Proof. We can assume that θ = 0. To prove the if-part, assume that 0 ∈ Θ for
some connected component Θ of the complement of C . We wish to show that f is
f
nevercolopsidedinΘ, forthisimpliesthatthecomplementofC hasVol(A)-many
f
connected components. Assume to the contrary that there exist a point θˆ ∈ Θ
such that f is colopsided at θˆ. Then, ord (f)(θˆ) = mπ for some integer m, with
B
|m| < Vol(A), see (7). Let fε = (f ,...,f ,f eiε). Then fε is colopsided at
0 n n+1
0 for ε ∈/ 2πZ. By continuity of roots, for ε > 0 sufficiently small, the points 0
and θˆare contained in the same connected component of the complement of C(fε).
Hence, by [7, pro. 3.9], they are contained in the same connected component of the
complement of L(fε). However,
ord (fε)(0)=Vol(A)(π−ε)(cid:54)=m(π−ε)=ord (fε)(θˆ),
B B
contradicting that ord (fε) is constant on connected components of the comple-
B
ment of L(fε).
To prove the only if-part, assume that there exists a connected component Θ
of the complement of C in which f is never colopsided. We wish to prove that
f
10 JENSFORSG˚ARD
0 ∈ Θ. As fε is colopsided at 0 for ε > 0 sufficiently small, we find that 0 ∈ Θ.
Indeed, if this was not the case, then the complement of C(fε) has (Vol(A)+1)-
many connected components, a contradiction. As 0 (cid:54)∈ H , and by [7, lem. 2.3],
f
there exists a disc D around 0 such that
0
D ∩(Tn\C )=D ∩Θ.
0 f 0
Furthermore, D ∩Θ(cid:54)=∅, since 0∈Θ. Let θ ∈D ∩Θ. As f is a real polynomial,
0 0
conjugation yields that −θ ∈ D ∩Θ. However, Θ ⊂ Rn is convex, implying that
0
0∈Θ. (cid:3)
Proof of Theorem 4.1. If Arg(f) (cid:54)∈ C then the image of ord (f) is of cardinality
∆ B
Vol(A), and hence f ∈ U . Thus, we only need to consider Arg(f) ∈ C , where
0 ∆
we can assume that fˆ(0) = δ for all k. In particular, f is a real polynomial. By
k
Proposition 4.5, it holds that the complement of C has Vol(A)-many connected
f
components if and only if it contains 0. Keeping f ,...,f and arg(f ) fixed, let
0 n n+1
us consider f as a function of |f |. As f is a real polynomial, it restricts to a
n+1
map f: Rn →R, whose image depends nontrivially on |f |. Notice that 0∈C
≥0 n+1 f
if and only if f(Rn ) contains the origin. Since fˆ(0) = δ = 1 for k (cid:54)= n+1, and
≥0 k k
sincea isaninteriorpointofA,themapf takestheboundaryofRn to[1,∞).
n+1 ≥0
In particular, if 0∈f(Rn ), then 0∈f(Rn). The boundary of the set of all |f |
≥0 + n+1
for which 0 ∈ f(Rn ) is the set of all values of |f | for which f(Rn) = [0,∞).
≥0 n+1 +
Furthermore, f(Rn)=[0,∞) holds if and only if there exists an r ∈Rn such that
+ +
f(r)=0, while f(r)≥0 in a neighborhood of r, implying that r is a critical point
of f. That is,
(cid:0) (cid:1)
∆ δ |f |,...,δ |f | =0.
1 1 n+1 n+1
Since ∆ is a binomial, there is exactly one such value of |f |. Finally, we note
n+1
that 0∈C if |f |→∞, which finishes the proof. (cid:3)
f n+1
4.2. Proof of Theorem 4.2. If Arg(f) (cid:54)∈ C , then the image of ord (f) is of
∆ B
cardinality Vol(A) and hence f ∈U . Assume that Arg(f)∈C , and that fˆ(0)=
0 ∆ k
δ for all k. It holds that 0∈H since there exists two adjacent vertices a and a
k f 0 1
of A such that δ =1 and δ =−1. Let,
0 1
H ={θ|(cid:104)a −a ,θ(cid:105)=0}
0 1
be the hyperplane of H containing 0. Assume that exists connected component
f
Θ of the complement of C in which f is nowhere colopsided. As in the proof
f
of Proposition 4.5, we conclude that 0 ∈ Θ, for otherwise we could construct a
coamoeba with (Vol(A)+1)-many connected components of its complement. As
H ⊂C , we find that Θ is contained in one of the half-spaces
f
H ={θ| ±(cid:104)a −a ,θ(cid:105)>0},
± 0 1
saythatΘ⊂H . Letfε =(f eiε,f ,...,f ),andletHε denotethecorrespond-
+ 0 1 n+1
ing hyperplane
Hε ={θ|(cid:104)a −a ,θ(cid:105)=−ε}.
0 1
For |ε| sufficiently small, continuity of roots implies that there is a connected com-
ponent Θε ⊂Hε in which fε is never colopsided. However, by choosing the sign of
+