In the tropical semifield we can also define an analogue of a polynomial. In this section we discuss this notion and the polyhedral complexes that arise as solutions to tropical polynomial inequalities.

**Definition 2.44.** A *(signed) tropical polynomial* over the variables *X*1*, . . . , X**n* is a formal
expression of the form

*P*(X) =^{⊕}

*α**∈**Λ*

*a*_{α}*⊙X*_{1}^{⊙}^{α}^{1}*⊙ · · · ⊙X*_{n}^{⊙}^{α}^{n}*,* (2.6)
where*Λ*is a finite subset of*{*0,1,2, . . .*}** ^{n}*, and

*a*

_{α}*∈*T

_{±}*\{−∞}*for all

*α∈Λ. Since the addition*inT

*is only partially defined, we cannot evaluate*

_{±}*P*(x) for all the points

*x∈*T

^{n}*. However, if*

_{±}*x∈*T

^{n}*is such that all of the terms*

_{±}*a*

*α*

*⊙x*

^{⊙}_{1}

^{α}^{1}

*⊙ · · · ⊙x*

^{⊙}

_{n}

^{α}*have the same sign, then we define*

^{n}*P*(x) as the tropical sum of these terms.

*Example*2.45. If*P*(X) = 2*⊙X*_{1}^{⊙}^{3}*⊙X*_{2}^{⊙}^{4}*⊕*(*⊖*0*⊙X*_{2}), then*P*(1,*⊖*5) = 25,*P(⊖*1,*−*5) =*⊖*(*−*5),
whereas*P* is not defined for (1,*−5/3).*

The next definition and lemma shows that a nonzero tropical polynomial divides R* ^{n}* into
cells that form a polyhedral complex. The union of the boundaries of cells of this complex is
called a

*tropical hypersurface—we refer to [MS15, Section 3.1] for more information.*

**Definition 2.46.** Given a tropical polynomial*P* as in (2.6), we say that*P* is*nonzero* if the set
*Λ* is nonempty. For every such tropical polynomial and every point*x∈*R* ^{n}*we define the

*set of*

*maximizing multi-indices at*

*x*as

Argmax(P, x) :=^{{}*α∈Λ*:*∀β* *∈Λ,* *|a**α**|*+*⟨α, x⟩*⩾*|a*_{β}*|*+*⟨β, x⟩*^{}}*,*
where*⟨·,·⟩* refers to the usual scalar product.

**Lemma 2.47.** *IfP* *is a nonzero tropical polynomial as in* (2.6)*and we fix a multi-indexα∈Λ,*
*then the set*

*W**α* := cl_{R}({x*∈*R* ^{n}*: Argmax(P, x) =

*α})*

*is a polyhedron that is either empty or full dimensional. Furthermore, if* *L* *⊂* *Λ* *is such that*
*α∈L, then the set*

*W**L*:= cl_{R}({x*∈*R* ^{n}*: Argmax(P, x) =

*L})*(2.7)

*is a (possibly empty) face of*

*W*

*α*

*. Moreover, the family{W*

*L*

*}*

*L*

*⊂*

*Λ*

*is a polyhedral complex whose*

*support is equal to*R

^{n}*. (Here,*cl

_{R}(

*·*)

*refers to the closure in*R

^{n}*and we use the convention that*

*W*

*=*

_{∅}*∅.)*

*Sketch of the proof.* To prove the first statement, note that a point*x∈*R* ^{n}*satisfies the equality
Argmax(P, x) =

*α*if and only if the inequality

*|a*

*α*

*|*+

*⟨α, x⟩*

*>*

*|a*

_{β}*|*+

*⟨β, x⟩*is true for all

*β*

*∈Λ\ {α}*. Therefore, if

*W*

*α*is nonempty, then it contains a ball. A relative interior of a face of

*W*

*α*is obtained by fixing a subset of inequalities of the form

*|a*

_{α}*|*+

*⟨α, x⟩>|a*

_{β}*|*+

*⟨β, x⟩*and

2.4. Tropical polynomials 27

Figure 2.1: Tropical hypersurface.

turning them into equalities. In other words, *V* is a (possibly empty) face of *W**α* if and only if
*V* = cl_{R}(*{x* *∈* R* ^{n}*: Argmax(P, x) =

*L}*) for some

*L*

*⊂Λ*such that

*α*

*∈L. The polyhedra*

*W*

*L*

have disjoint relative interiors, and hence the family *{W**L**}**L**⊂**Λ* is a polyhedral complex. The
fact that its support is equal to R* ^{n}* follows from the definition.

*Example* 2.48. Take

*P*(X_{1}*, X*_{2}) :=^{(}2*⊙X*_{1}^{)}*⊕*^{(}*⊖*(*−*4)*⊙X*_{1}^{⊙}^{2})^{)}*⊕*^{(}*⊖*(*−*3)*⊙X*_{1}*⊙X*_{2}^{)}*⊕*^{(}*⊖*5^{)}*.*

The tropical hypersurface associated with *P* is depicted in Fig. 2.1. The associated polyhedral
complex consists of 11 nonempty polyhedra, 4 of which are full dimensional (one for every
monomial of*P*).

In this work, we are interested in systems of tropical inequalities. To do so, it is convenient to make the following definitions.

**Definition 2.49.** Given a tropical polynomial*P* as in (2.6), we set *Λ*^{+} := *{α* *∈Λ*:*a*_{α}*∈*T+*}*
and *Λ** ^{−}* :=

*{α*

*∈Λ*:

*a*

_{α}*∈*T

_{−}*}*. Furthermore, we define

*P*

^{+}:=

^{⊕}

*+*

_{α∈Λ}*a*

_{α}*⊙X*

_{1}

^{⊙}

^{α}^{1}

*⊙ · · · ⊙X*

_{n}

^{⊙}

^{α}*and*

^{n}*P*

*:=*

^{−}^{⊕}

_{α}

_{∈}

_{Λ}*−*

*|a*

*α*

*| ⊙X*

_{1}

^{⊙}

^{α}^{1}

*⊙ · · · ⊙X*

_{n}

^{⊙}

^{α}*. Finally, if*

^{n}*P*is nonzero, then we set

S^{⩾}(P) :=*{x∈*R* ^{n}*:

*P*

^{+}(x)⩾

*P*

*(x)*

^{−}*}*

(with the convention that*P*^{+}(x) :=*−∞*if*Λ*^{+}=*∅* and *P** ^{−}*(x) :=

*−∞*if

*Λ*

*=*

^{−}*∅).*

*Remark* 2.50. We point out the values*P*^{+}(x), P* ^{−}*(x) are well defined for all

*x∈*T

*. Therefore, S*

^{n}^{⩾}(P) is also well defined.

The following definition and a lemma show the connection between polynomial inequalities
overK_{⩾}0 and tropical polynomial inequalities.

**Definition 2.51.** To any polynomial
* P*(X) =

^{∑}

*α**∈**Λ*

**a***α**X*_{1}^{α}^{1}*. . . X*_{n}^{α}^{n}*∈*K[X1*, . . . , X**n*] (2.8)
over Puiseux series we associate a tropical polynomial defined as

trop(P) := ^{⊕}

*α**∈**Λ*

sval(a*α*)*⊙X*_{1}^{⊙}^{α}^{1}*⊙ · · · ⊙X*_{n}^{⊙}^{α}^{n}*.*

28 Chapter 2. Preliminaries

Figure 2.2: Support of the polyhedral complex*C*^{⩾}(P).

**Lemma 2.52.** *Let* **P***∈* K[X_{1}*, . . . , X** _{n}*]

*be a polynomial and let*

*P*:= trop(P). If

*K*

**x**∈

^{n}_{⩾}0

*is*

*such that*

*(x)⩾0, then*

**P***P*

^{+}

^{(}val(x)

^{)}⩾

*P*

^{−}^{(}val(x)

^{)}

*.*

*Proof.* Define *x*=val(x)*∈*T^{n}* _{±}* and let

**P**^{+}(X) = ^{∑}

*α**∈**Λ*^{+}

**a***α**X*_{1}^{α}^{1}*. . . X*_{n}^{α}^{n}**P*** ^{−}*(X) =

^{∑}

*α**∈**Λ*^{−}

*|a**α**|X*_{1}^{α}^{1}*. . . X*_{n}^{α}^{n}*.*

By (2.4) and Remark 2.36, we have *P*^{+}^{(}val(x)^{)}=val^{(}**P**^{+}(x)^{)} and *P*^{−}^{(}val(x)^{)} =val^{(}**P*** ^{−}*(x)

^{)}. Moreover,

**P**^{+}(x)⩾

**P***(x) and the claim follows from (2.5).*

^{−}Next, we define the signed version of the polyhedral complex associated with a tropical polynomial and show its connections to tropical polynomial inequalities.

**Definition 2.53.** We say that a cell *W**L* *∈ C*(P) as in (2.7) is *positive* if there exists at least
one*α∈L*such that*a**α* *∈*T+ or if*W**L* is empty.

**Lemma 2.54.** *Suppose that* *P* *is a nonzero tropical polynomial and let* *C*^{⩾}(P) *be the family of*
*positive cells of* *C*(P). Then, *C*^{⩾}(P) *is a polyhedral complex whose support is equal to*S^{⩾}(P).

*Sketch of the proof.* As in the proof of Lemma 2.47, if *W**L* belongs to *C*^{⩾}(P) and *V* is its face,
then *V* =*W**L** ^{′}* for some

*L*

*⊂L*

*. This means that all faces of*

^{′}*W*

*L*belong to

*C*

^{⩾}(P). Moreover, the polyhedra in

*C*

^{⩾}(P) have disjoint relative interiors and so

*C*

^{⩾}(P) is a polyhedral complex.

To finish the proof, observe that*x∈*S^{⩾}(P) if and only if Argmax(P, x)*∩Λ*^{+}*̸*=*∅*.

*Example* 2.55. Let *P*(X1*, X*2) be as in Example 2.48. Then, the polyhedral complex *C*^{⩾}(P)
consists of 6 polyhedra and its support is depicted in Fig. 2.2.

Given a system of nonzero tropical polynomials *P*_{1}*, . . . , P** _{m}*, we also regard the refinements
of complexes defined by

*P*1

*, . . . , P*

*m*. More precisely, we make the following definition.

**Definition 2.56.** If*P*_{1}*, . . . , P** _{m}* are nonzero tropical polynomials, then we define

*C*(P

_{1}

*, . . . , P*

*) and*

_{m}*C*

^{⩾}(P

_{1}

*, . . . , P*

*) as*

_{m}*C*(P_{1}*, . . . , P** _{m}*) :=

^{{}

*∩*

^{m}*i=1*

*V*

*i*:

*∀i,*

*V*

*i*

*∈ C*(P

*)*

_{i}^{}}

*,*

*C*

^{⩾}(P1

*, . . . , P*

*m*) :=

^{{}

*∩*

^{m}

_{i=1}*V*

*i*:

*∀i,*

*V*

*i*

*∈ C*

^{⩾}(P

*i*)

^{}}

*.*

2.4. Tropical polynomials 29
**Lemma 2.57.** *The families* *C*(P1*, . . . , P**m*) *and* *C*^{⩾}(P1*, . . . , P**m*) *are polyhedral complexes. The*
*support of the former is equal to* R^{n}*while the support of the latter coincides with*

S^{⩾}(P1*, . . . , P**m*) :=*{x∈*R* ^{n}*:

*∀i, P*

_{i}^{+}(x)⩾

*P*

_{i}*(x)*

^{−}*}.*

*Proof.*Obvious from Lemma 2.9 and the lemmas above.

Finally, in this work we often consider polyhedral complexes with regular supports. Recall
that a closed set *X* *⊂* R* ^{n}* is called

*regular*if

*X*= cl

_{R}(int(X)). The next lemmas show basic properties of these complexes.

**Lemma 2.58.** *If* *Cis a polyhedral complex, then its support is regular if and only if every point*
*of the support belongs to some full-dimensional cell of* *C.*

*Proof.* Let S denote the support of *C. If* *x* *∈*S belongs to some full-dimensional cell*W* of *C,*
then *x* *∈ W* = cl_{R}(int(*W*)) *⊂* cl_{R}(int(S)) by Lemma 2.3. This proves the “if” direction. To
prove the opposite direction, suppose thatSis regular and take any point *y∈*int(S). We will
show that *y* belongs to some full-dimensional cell of *C. Since the polyhedra that form* *C* are
closed (and there are finitely many of them), there exists an open ball*B*(y, r)*⊂*R* ^{n}* such that
a cell

*W ∈ C*has a nonempty intersection with

*B(y, r)*if and only if

*y*

*∈ W*. Let 0

*< r*

^{′}*< r*be such that

*B(y, r*

*)*

^{′}*⊂*S. If

*y*does not belong to a full-dimensional cell, then the union of all cells containing

*y*is not full dimensional and it does not contain a ball (by Proposition 2.20).

In particular, *B(y, r** ^{′}*) is not contained in this union, which gives a contradiction with the fact
that

*r*

^{′}*< r. Hence,y*belongs to some full-dimensional cell of

*C.*

To finish the proof, take any *x* *∈* S. Then for every *k* ⩾ 1, there exists a point *y*^{(k)} *∈*
int(S)*∩B(x,*1/k). By the observation above, *y*^{(k)} belongs to some full-dimensional cell of
*C*. Therefore, up to taking a subsequence, we can suppose that all *y*^{(k)} belong to the same
full-dimensional cell *W ∈ C*. Since*W* is closed, we get *x∈ W*.

**Lemma 2.59.** *Suppose that the polyhedral complexC*^{⩾}(P1*, . . . , P**m*)*has a regular support. Then*
*this support,* S^{⩾}(P1*, . . . , P**m*), coincides with the closure of the set

S* ^{>}*(P1

*, . . . , P*

*m*) :=

*{x∈*R

*:*

^{n}*∀i, P*

_{i}^{+}(x)

*> P*

_{i}*(x)}*

^{−}*.*

*Proof.* If there is a tropical polynomial*P** _{i}* such that

*P*=

*P*

_{i}*, then both of these sets are empty and the claim is trivially true. If there is a tropical polynomial*

^{−}*P*

*i*such that

*P*=

*P*

_{i}^{+}, then we can remove this polynomial from the collection and this does not change the sets S

^{⩾}(P1

*, . . . , P*

*m*) and S

*(P*

^{>}_{1}

*, . . . , P*

*). Hence, we can suppose that all the tropical polynomials*

_{m}*P*

_{i}^{+}

*, P*

_{i}*are nonzero. Then, the setS*

^{−}^{⩾}(P1

*, . . . , P*

*m*)is closed since the tropical polynomial functions

*P*

_{i}*are continuous. We obviously haveS*

^{±}*(P1*

^{>}*, . . . , P*

*m*)

*⊂*S

^{⩾}(P1

*, . . . , P*

*m*). Therefore,

cl_{R}(S* ^{>}*(P1

*, . . . , P*

*m*))

*⊂*S

^{⩾}(P1

*, . . . , P*

*m*)

*.*

Consider now*y∈*S^{⩾}(P_{1}*, . . . , P** _{m}*). Since this set is regular, by Lemma 2.58,

*y*belongs to a full-dimensional cell

*W*of

*C*

^{⩾}(P1

*, . . . , P*

*m*). We have

*W*=

*∩*1⩽

*i*⩽

*m*

*W*

*i*, where

*W*

*i*is a full-dimensional cell of

*C*

^{⩾}(P

*i*). This implies that

*W*

*i*= cl

_{R}(

*{x*

*∈*R

*: Argmax(P*

^{n}*i*

*, x) =*

*L*

*i*

*}*) where

*L*

*i*is a one element subset of

*Λ*

^{+}(P

*). We conclude that*

_{i}*P*

_{i}^{+}(x)

*> P*

_{i}*(x) holds for all*

^{−}*x*

*∈*int(

*W*

*i*), and so, int(W)

*⊂*S

*(P1*

^{>}*, . . . , P*

*m*). We have

*W*= cl

_{R}(int(W)) by Lemma 2.3 and hence

*y∈*cl

_{R}(S

*(P1*

^{>}*, . . . , P*

*m*)).

30 Chapter 2. Preliminaries