We perform the first transformation on the graph *G⃗* and denote the transformed graph by *G⃗*1.
We then perform the second transformation on every edge in*G⃗*1that joins two Random vertices.

We denote the transformed graph by*G⃗** ^{′}* and the associated operator as

*F*

*:R*

^{′}

^{n+n}

^{′}*→*R

^{n+n}*. By Lemmas 5.42 and 5.43, the real tropical cone*

^{′}*{x∈*R

*:*

^{n}*x*⩽

*F*(x)

*}*is the projection of the real tropical cone

*{*(x, x

*)*

^{′}*∈*R

^{n}*×*R

^{n}*: (x, x*

^{′}*)⩽*

^{′}*F*

*(x, x*

^{′}*)*

^{′}*}*. Furthermore,

*G⃗*

*fulfills the conditions of Proposition 5.40. Therefore, the set S*

^{′}*=*

^{′}*{(x, x*

*)*

^{′}*∈*T

^{n}*×*T

^{n}*: (x, x*

^{′}*) ⩽*

^{′}*F*

*(x, x*

^{′}*)} is a tropical Metzler spectrahedral cone. Finally, we take the set*

^{′}S* ^{′′}*=

*{(x, x*

^{′}*, y)∈*T

^{n}*×*T

^{n}

^{′}*×*T

^{n+n}*: (x, x*

^{′}*)⩽*

^{′}*F*

*(x, x*

^{′}*)*

^{′}*∧*(x, x

*) +*

^{′}*y*⩾0}

=*{(x, x*^{′}*, y)∈*R^{n}*×*R^{n}^{′}*×*R^{n+n}* ^{′}*: (x, x

*)⩽*

^{′}*F*

*(x, x*

^{′}*)*

^{′}*∧*(x, x

*) +*

^{′}*y*⩾0}

*.*

The setS* ^{′′}* is a real tropical Metzler spectrahedron because a constraint of the form

*x*

*v*+

*y*

*v*⩾0 can be encoded by adding a2

*×*2block

[*x*_{v}*⊖*0

*⊖0* *y**v*

]

to the matrices that describeS* ^{′}*. Moreover,Sis a projection of S

*. Indeed, if(x, x*

^{′′}*) is a real vector such that (x, x*

^{′}*)⩽*

^{′}*F*

*(x, x*

^{′}*), then there exists*

^{′}*y*

*∈*R

^{n+n}*such that (x, x*

^{′}*) +*

^{′}*y*⩾0.

*Example* 5.45. Take the graph from Fig. 5.3 and consider the Random vertex that has Min
vertices 2 and 3 as its neighbors. Figure 5.8 presents the outcome of the procedure described
in the lemmas above when applied to this vertex.

**5.5** **Tropical Helton–Nie conjecture for Puiseux series**

We now generalize Proposition 5.44 to tropically convex sets inT* ^{n}*. In order to study this case,
we use the notion of homogenization of a convex set. There are many possible homogenizations
of a given set and we need to use three of them.

**Definition 5.46.** IfSis a tropically convex set with only finite points (i.e.,S_{⊂}_{R}* ^{n}*), then we
define its

*real homogenization*as

S* ^{rh}* :=

*{*(x

_{0}

*, x*

_{0}+

*x)∈*R

*:*

^{n+1}*x∈*S

*IfS*

_{}}.

_{⊂}_{T}

*is a tropically convex set, then we define its*

^{n}*homogenization*as

S* ^{h}*:=

*{*(x

_{0}

*, x*

_{0}+

*x)∈*T

*:*

^{n+1}*x∈*S

_{}}.If *S(Q*^{(0)}*|Q*^{(1)}*, . . . , Q*^{(n)}) *⊂* T* ^{n}* is a tropical Metzler spectrahedron, then we define its

*formal*

*homogenization*

*S*

^{f h}*⊂*T

*as the tropical Metzler spectrahedral cone*

^{n+1}*S** ^{f h}*:=

*S*(Q

^{(0)}

*, Q*

^{(1)}

*, . . . , Q*

^{(n)})

*.*

**Lemma 5.47.** *If* S _{⊂}_{T}^{n}*is tropically convex, then its homogenization is a tropical cone.*

*Moreover, if* S*is included in* R^{n}*, then its real homogenization if a real tropical cone.*

100 Chapter 5. Tropical analogue of the Helton–Nie conjecture

Figure 5.8: The transformation of Lemmas 5.41 to 5.43 applied to one Random vertex from the graph presented in Fig. 5.3. Top left: the initial graph. Top right: the graph after the application of Lemma 5.41. Bottom left: the graph after the application of Lemma 5.42. Bottom right: the graph after the application of Lemma 5.43.

*Proof.* Suppose that S _{⊂}_{T}* ^{n}* is tropically convex, take its homogenization S

*, two points (x*

^{h}_{0}

*, x*

_{0}+

*x),*(y

_{0}

*, y*

_{0}+

*y)∈*S

*(such that*

^{rh}*x, y∈*Sand

*x*

_{0}

*, y*

_{0}

*∈*T) and

*λ, µ∈*T. Without loss of The proof for the real homogenization is analogous.

*Example* 5.48. The three notions of homogenization given in Definition 5.46 are diﬀerent.
In-deed, if we take the set S := *{x* *∈* T: *x* ⩾ 0*}*, then its real homogenization if equal to
S* ^{rh}* =

*{*(x, y)

*∈*R

^{2}:

*y*⩾

*x}*, while its homogenization is equal to S

*= S*

^{h}

^{rh}*. More-over, S=*

_{∪ {−∞}}*S(Q*

^{(0)}

*|Q*

^{(1)}) is a tropical Metzler spectrahedron defined by the matrices

*Q*

^{(0)}=

*⊖0,*

*Q*

^{(1)}= 0. Then, its formal homogenization is equal to

*S*

*=*

^{f h}*{*(x, y)

*∈*T

^{2}:

*y*⩾

*x}*.

**Lemma 5.49.** *Every closed, semilinear, tropically convex set in* R^{n}*is a projection of a real*
*tropical Metzler spectrahedron.*

*Proof.* Take any closed, semilinear, tropically convex set S_{⊂}_{R}* ^{n}* and consider its real
homog-enizationS

*. This set is a real tropical cone in R*

^{rh}*by Lemma 5.47. Moreover, it is closed*

^{n+1}5.5. Tropical Helton–Nie conjecture for Puiseux series 101
and definable in *L*og. This implies that it is semilinear by Lemma 2.103. By Proposition 5.44,
S* ^{rh}* is a projection of a real tropical Metzler spectrahedron

*S*1

*⊂*R

*×*R

^{n}*×*R

^{n}*. Consider the set*

^{′}*S*2=*{*(x0*, x, y)∈*R*×*R^{n}*×*R^{n}* ^{′}*: (x0

*, x, y)∈ S*1

*∧x*0= 0

*}.*

The set *S*2 is a tropical Metzler spectrahedron because the constraint *x*0 = 0 can be encoded
by adding a2*×*2 block _{[}

*⊖*0*⊕x*_{0} *−∞*

*−∞* 0*⊕*(*⊖x*_{0})
]

to the matrices that describe *S*1. Furthermore, S is its projection. Indeed, if *x* *∈* S, then
(0, x)*∈*S* ^{rh}* and there exists

*y*

*∈*R

^{n}*such that(0, x, y)*

^{′}*∈ S*1. Conversely, if(0, x, y)

*∈ S*1, then (0, x)

*∈*S

*and hence*

^{rh}*x∈*S.

We now want to extend the result of Lemma 5.49 to tropically convex sets in T* ^{n}*. In
order to do this, we proceed stratum-by-stratum. This requires to show that a tropical convex
hull of finitely many projected tropical Metzler spectrahedra is a projected tropical Metzler
spectrahedron. In the classical case of real spectrahedra, it is known that a convex hull of
finitely many projected spectrahedra is a projected spectrahedron. This fact has a very short
proof presented in [NS09]. The proof in the tropical case is exactly the same (we only change
the classical notation to the tropical one). Let us present this proof for the sake of completeness.

**Lemma 5.50.** *A tropically convex set* S_{⊂}_{T}^{n}*is a projected tropical Metzler spectrahedron if*
*and only if its homogenization is a projected tropical Metzler spectrahedron.*

*Proof.* First, suppose thatS* ^{h}* is a projection of a tropical Metzler spectrahedron

*S*1

*⊂*T

*×*T

^{n}*×*T

^{n}*. Consider the set*

^{′}*S*2 =*{*(x0*, x, y)∈*T*×*T^{n}*×*T^{n}* ^{′}*: (x0

*, x, y)∈ S*1

*∧x*0 = 0

*}.*

As in the proof of Lemma 5.49, the set *S*2 is a tropical Metzler spectrahedron and S is its
projection. Conversely, suppose that S is a projection of a tropical Metzler spectrahedron
*S*1 *⊂*T^{n}*×*T^{n}* ^{′}*. Consider its formal homogenization

*S*

_{1}

^{f h}*⊂*T

^{1+n+n}

*and take the set*

^{′}*S*2=*{*(x0*, x, y, z*)*∈*T*×*T^{n}*×*T^{n}^{′}*×*T* ^{n}*: (x0

*, x, y)∈ S*1

^{f h}*∧ ∀k∈*[n], x0+

*z*

*⩾2x*

_{k}

_{k}*}.*The set

*S*2 is a tropical Metzler spectrahedron because a constraint of the form

*x*0+

*z*

*k*⩾2x

*k*

can be encoded by adding a2*×*2block

[ *x*_{0} *⊖x*_{k}

*⊖x**k* *z**k*

]

to the matrices that describe *S*1. We will show that S* ^{h}* is a projection of

*S*2. To see this, take any point (x

_{0}

*, x*

_{0}+

*x)*

*∈*S

*, where*

^{h}*x*

*∈*S and

*x*

_{0}

*∈*T. If

*x*

_{0}=

*−∞*, then

*−∞ ∈*S

*belongs to the projection of*

^{h}*S*2. Otherwise, take

*y*such that (x, y)

*∈ S*1 and observe that we have (x0

*, x*0 +

*x, x*0 +

*y)*

*∈ S*

_{1}

*. Since*

^{f h}*x*0

*∈*R, if we take

*z*

*k*large enough, then (x0

*, x*0+

*x, x*0+

*y, z*)

*∈ S*2. This shows thatS

*is included in the projection of*

^{h}*S*2. Conversely, suppose that (x

_{0}

*, x, y, z)*

*∈ S*2. If

*x*

_{0}=

*−∞*, then

*x*=

*−∞*and hence (x

_{0}

*, x)*

*∈*S

*. If*

^{h}*x*

_{0}

*̸*=

*−∞*, then we have(0,

*−x*

_{0}+

*x,−x*

_{0}+

*y,−x*

_{0}+

*z)∈ S*2. Hence(0,

*−x*

_{0}+

*x,−x*

_{0}+

*y)∈ S*

_{1}

*, (*

^{f h}*−x*

_{0}+

*x,−x*

_{0}+

*y)∈ S*1, and

*−x*

_{0}+

*x∈*S. Therefore(x

_{0}

*, x)∈*S

*.*

^{h}102 Chapter 5. Tropical analogue of the Helton–Nie conjecture
**Lemma 5.51.** *Suppose that* S_{1}*,*S_{2} _{⊂}_{T}^{n}*are projected tropical Metzler spectrahedra. Then*
tconv(S_{1}* _{∪}*S

_{2})

*is a projected tropical Metzler spectrahedron.*

*Proof.* LetS= tconv(S_{1}* _{∪}*S

_{2})and consider

S_{1}^{h}* _{⊕}*S

_{2}

*:=*

^{h}*{x∈*T

*:*

^{n+1}*∃(u, w)∈*S

_{1}

^{h}*S*

_{×}_{2}

^{h}*, x*=

*u⊕w}.*

First, we will show that we have the identity S* ^{h}* = S

_{1}

^{h}*S*

_{⊕}_{2}

*. Indeed, since S*

^{h}_{1}

*S, we have S*

_{⊂}_{1}

^{h}*S*

_{⊂}*. Similarly, S*

^{h}_{2}

^{h}*S*

_{⊂}*. Therefore, we have S*

^{h}_{1}

^{h}*S*

_{⊕}_{2}

^{h}*S*

_{⊂}*. Conversely, take a point*

^{h}*z∈*S

*. By Lemma 5.17, we can write*

^{h}*z*as

*z*=
(

*z*0*, z*0*⊙*^{(}(λ*⊙x)⊕*(µ*⊙y)*^{))}*∈*S^{h}*,*
where*λ⊕µ*= 0,*x∈*S_{1}, and *y∈*S_{2}. Then *z*= ˜*x⊕y, where*˜

˜

*x*:= (λ*⊙z*0*,*(λ*⊙z*0)*⊙x)∈*S_{1}^{h}*,*

˜

*y*:= (µ*⊙z*0*,*(µ*⊙z*0)*⊙y)∈*S_{2}^{h}*.*

HenceS* ^{h}* =S

_{1}

^{h}*S*

_{⊕}_{2}

*. SinceS*

^{h}_{1}

*,*S

_{2}are projected tropical Metzler spectrahedra, the same is true for S

_{1}

^{h}*,*S

_{2}

*by Lemma 5.50. Let S*

^{h}_{1}

*be a projection of*

^{h}*S*1

*⊂*T

^{n+1+n}^{1}and S

_{2}

*be a projection of*

^{h}*S*2

*⊂*T

^{n+1+n}^{2}. Then, the set S

_{1}

^{h}*S*

_{×}_{2}

*is a projection of the set*

^{h}*S*1

*× S*2. Moreover, the set

*S*1

*× S*2 is a tropical Metzler spectrahedron (described by block-diagonal matrices such that the first block is given by the matrices that describe

*S*1 and the second block is given by the matrices that describe

*S*2). Therefore, the set

*S*3:=*{*(u, u^{′}*, w, w*^{′}*, x) :* T^{(n+1)+n}^{1}^{+(n+1)+n}^{2}^{+(n+1)}: (u, u^{′}*, w, w** ^{′}*)

*∈ S*1

*× S*2

*, x*=

*u⊕w}*is a tropical Metzler spectrahedron (because a constraint of the form

*x*

*=*

_{k}*u*

_{k}*⊕w*

*can be encoded by adding a2*

_{k}*×*2block to the matrices that describe

*S*1

*× S*2). Moreover,S

_{1}

^{h}*S*

_{⊕}_{2}

*is a projection of*

^{h}*S*3. Since S

*=S*

^{h}_{1}

^{h}*S*

_{⊕}_{2}

*, the set Sis a projected tropical Metzler spectrahedron by Lemma 5.50.*

^{h}We are now ready to present the proof of Theorem 5.5.

*Proof of Theorem 5.5.* The equivalence between Theorem 5.5 (a) and Theorem 5.5 (b) is given
in Proposition 5.21. The implication from Theorem 5.5 (b) to Theorem 5.5 (c) follows from
Lemma 5.49. We now prove the implication from Theorem 5.5 (c) to Theorem 5.5 (d). Let
S _{⊂}_{T}* ^{n}* be as in Theorem 5.5 (c). If S is empty, then it is a tropical Metzler spectrahedron
defined by a single inequality

*−∞*⩾ 0. Otherwise, let

*K*

*⊂*[n] be any nonempty set such that the stratumS

_{K}

_{⊂}_{R}

*is nonempty. The setS*

^{K}*is a projection of a real tropical Metzler spectrahedron*

_{K}*S*

*K*

*⊂*R

^{K}*×*R

^{n}*. For any*

^{′}*x∈*T

*we denote by*

^{n}*x*

_{K}*∈*T

*the subvector formed by the coordinates of*

^{K}*x*with indices in

*K. Furthermore, let*

*X*

*K*

*⊂*T

*denote the set*

^{n}*X**K* :=*{x∈*T* ^{n}*:

*x*

*k*

*̸*=

*−∞ ⇐⇒*

*k∈K}.*

The setS_{∩}X*K* is a projection of a tropical Metzler spectrahedron defined as
*S*˜*K* =*{*(x, y)*∈*T^{n}*×*T^{n}* ^{′}*: (x

_{K}*, y)∈ S*

*K*

*∧ ∀k /∈K,−∞*⩾

*x*

_{k}*}.*

Moreover, for*K*=*∅*, let us denote*X** _{∅}* =

*−∞*. Note that the intersectionS

_{∩}X*is either empty or is equal to*

_{∅}*−∞*, and that

*−∞*is a tropical Metzler spectrahedron (defined by the inequalities