In this section we show that the requirements of Theorems 4.19 and 4.28 on the matrices*Q*^{(k)}
and the regularity of sets are fulfilled generically. In [ABGJ15] it was shown that genericity
conditions for tropical polyhedra can be described by the means of tangent digraphs. We extend
this characterization to tropical spectrahedra. For this purpose, we work with hypergraphs
instead of graphs.

**Definition 4.33.** A*(directed) hypergraph* is a pair*G⃗*:= (V, E), where*V* is a finite set of*vertices*
and*E* is a finite set of*(hyper)edges. Every edgee∈E* is a pair(T_{e}*, h** _{e}*), where

*h*

_{e}*∈V*is called the

*head*of the edge, and

*T*

*is a multiset with elements taken from*

_{e}*V*. We call

*T*

*the*

_{e}*multiset*

*of tails ofe. By|T*

*e*

*|*we denote the cardinality of

*T*

*e*(counting multiplicities). We do not exclude the situation in which a head is also a tail, i.e., it is possible that

*h*

_{e}*∈T*

*. If*

_{e}*v∈V*is a vertex of

*G⃗*, then by In(v)

*⊂E*we denote the set of

*incoming edges, i.e., the set of all edges*

*e*such that

*h*

*e*=

*v. By*Out(v)we denote the multiset of

*outgoing edges, i.e., a multiset of edgese*such that

*v*

*∈T*

*. We treatOut(v) as a multiset, with the convention that*

_{e}*e∈E*appears

*p*times in Out(v) if

*v*appears

*p*times in

*T*

*.*

_{e}Let us now define the notion of a circulation in a hypergraph.

**Definition 4.34.** A *circulation* in a hypergraph is a vector *γ* = (γ* _{e}*)

_{e}

_{∈}*such that*

_{E}*γ*

*⩾0 for all*

_{e}*e∈E,*

^{∑}

_{e}

_{∈}

_{E}*γ*

*e*= 1, and for all

*v∈V*we have the equality

∑

*e**∈*In(v)

*|T**e**|γ**e*= ^{∑}

*e**∈*Out(v)

*γ**e**.*

Observe that if a hypergraph*G⃗* is fixed, then the set of all normalized circulations on*G⃗* forms a
polyhedron. We say that a hypergraph*does not admit a circulation* if this polyhedron is empty.

In this section, we only consider hypergraphs such that every edge has at most two tails
(counting multiplicities). Hereafter,*ϵ** _{k}* denotes the

*kth vector of the standard basis in*R

*.*

^{n}**Definition 4.35.**Given a sequence of tropical symmetric Metzler matrices

*Q*

^{(1)}

*, . . . , Q*

^{(n)}

*∈*T

^{m×m}*and a point*

_{±}*x*

*∈*R

*, we construct a hypergraph associated with*

^{n}*x, denoted*

*G⃗*

*x*, as follows:

• we put*V* := [n];

• for every *i* *∈*[m] verifying *Q*^{+}* _{ii}*(x) =

*Q*

^{−}*(x)*

_{ii}*̸*=

*−∞*, and every pair

*ϵ*

_{k}*∈*Argmax(Q

^{+}

_{ii}*, x),*

*ϵ*

_{l}*∈*Argmax(Q

^{−}

_{ii}*, x), the hypergraphG⃗*

*x*contains an edge(k, l);

• for every *i < j* such that *Q*^{+}* _{ii}*(x)

*⊙Q*

^{+}

*(x) = (Q*

_{jj}*(x))*

_{ij}

^{⊙}^{2}

*̸*=

*−∞*and every triple

*ϵ*

_{k}_{1}

*∈*Argmax(Q

^{+}

_{ii}*, x),*

*ϵ*

_{k}_{2}

*∈*Argmax(Q

^{+}

_{jj}*, x),*

*ϵ*

_{l}*∈*Argmax(Q

_{ij}*, x), the hypergraph*

*G⃗*

*x*contains an edge(

*{k*1

*, k*2

*}, l).*

The next lemma shows that the regularity assumptions given in Theorem 4.19 are automat-ically satisfied if the hypergraphs constructed in Definition 4.35 do not admit a circulation. The proof uses Farkas’ lemma.

4.3. Genericity conditions 75
**Theorem 4.36** (Farkas’ lemma, [Sch87, Corollary 7.1d]). *The polyhedronW* =*{x∈*R^{n}_{⩾}0:*∀i∈*
[p],*⟨a*_{i}*, x⟩*=*b*_{i}*}* *is empty if and only if there exists a vectory∈*R^{p}*such that*^{∑}^{p}_{i=1}*b*_{i}*y*_{i}*<*0 *and*

∑*p*

*i=1**a*_{ik}*y** _{i}* ⩾0

*for all*

*k∈*[n].

**Lemma 4.37.** *Suppose that for every* *x∈*R^{n}*the hypergraphG⃗**x* *does not admit a circulation.*

*Then, the matrices* *Q*^{(1)}*, . . . , Q*^{(n)} *fulfill Assumption A and* *S(Q*^{(1)}*, . . . , Q*^{(n)})*∩*R^{n}*is regular.*

*Proof.* To prove the first part, suppose that we have *Q*^{(k)}* _{ii}* +

*Q*

^{(k)}

*= 2|Q*

_{jj}^{(k)}

_{ij}*|*for some

*i̸=j*and

*Q*

^{(k)}

_{ii}*, Q*

^{(k)}

_{jj}*∈*T+. Take the point

*x*:=

*N ϵ*

*k*

*∈*R

*. If*

^{n}*N*is large enough, then we have

*Q*^{+}* _{ii}*(x)

*⊙Q*

^{+}

*(x) =*

_{jj}*Q*

^{(k)}

*+*

_{ii}*Q*

^{(k)}

*+ 2N = 2*

_{jj}*|Q*

^{(k)}

_{ij}*|*+ 2N = (Q

*(x))*

_{ij}

^{⊙}^{2}

and the hypergraph*G⃗**x* contains the edge(*{k, k}, k). This hypergraph admits a circulation (we*
put*γ**e* := 1for*e*= (*{k, k}, k)* and *γ**e*:= 0 for other edges), which gives a contradiction.

We now claim that the set *S ∩*R* ^{n}* is regular. Let

*T*be defined as in Definition 4.16. Let us show that for every

*x*

*∈ S ∩*R

*there exists a vector*

^{n}*η*

*∈*R

*such that*

^{n}*x*+

*ρη*belongs to

*T*for

*ρ >*0 small enough. This is suﬃcient to prove the claim because

*T ∩*R

*is a subset of the interior of*

^{n}*S ∩*R

*(as shown in the proof of Theorem 4.19). Fix a point*

^{n}*x*

*∈ S ∩*R

*. If*

^{n}*x*belongs to

*T*, then we can take

*η*:= 0. Otherwise, let

*G⃗*

*x*denote the hypergraph associated with

*x. The polytope of circulations of this hypergraph is empty. Therefore, by Farkas’ lemma*(Theorem 4.36), there exists a vector

*η∈*R

*such that for every edge*

^{n}*e∈E*we have

∑

*v**∈**T**e*

*η*_{v}*>|T*_{e}*|η*_{h}_{e}*.*

Take the vector *x*^{(ρ)}:=*x*+*ρη. Let us look at two cases.*

First, suppose that there is *i∈*[m]such that*Q*^{+}* _{ii}*(x) =

*Q*

^{−}*(x)*

_{ii}*̸*=

*−∞*. Fix any

*k*

*such that*

^{∗}*ϵ*

*k*

^{∗}*∈*Argmax(Q

^{+}

_{ii}*, x)*and take any

*l*such that

*ϵ*

*l*

*∈*Argmax(Q

^{−}

_{ii}*, x). Then*(k

^{∗}*, l)*is an edge in

*G⃗*

*x*. Therefore

*η*

_{k}*∗*

*> η*

*. Moreover,*

_{l}*Q*

^{(k}

_{ii}

^{∗}^{)}+

*x*

_{k}*∗*=

*|Q*

^{(l)}

_{ii}*|*+

*x*

*and hence*

_{l}*Q*

^{(k}

_{ii}

^{∗}^{)}+

*x*

^{(ρ)}

_{k}

_{∗}*>|Q*

^{(l)}

_{ii}*|*+

*x*

^{(ρ)}

*. Furthermore, for every*

_{l}*l*

^{′}*∈/*Argmax(Q

^{−}

_{ii}*, x)*we have

*Q*^{(k}_{ii}^{∗}^{)}+*x*_{k}*∗* =*|Q*^{(l)}_{ii}*|*+*x*_{l}*>|Q*^{(l}_{ii}^{′}^{)}*|*+*x*_{l}*′**.*

Therefore *Q*^{(k}_{ii}^{∗}^{)}+*x*^{(ρ)}_{k}_{∗}*>|Q*^{(l}_{ii}^{′}^{)}*|*+*x*^{(ρ)}_{l}* _{′}* for

*ρ*small enough. Since

*l,*

*l*

*were arbitrary, for every suﬃciently small*

^{′}*ρ*we have

*Q*^{+}* _{ii}*(x

^{(ρ)})⩾

*Q*

^{(k}

_{ii}

^{∗}^{)}+

*x*

^{(ρ)}

_{k}

_{∗}*> Q*

^{−}*(x*

_{ii}^{(ρ)})

*.*

The second case is analogous. If there is*i < j*such that*Q*^{+}* _{ii}*(x)⊙Q

^{+}

*(x) = (Q*

_{jj}*ij*(x))

^{⊙}^{2}

*̸=−∞,*then we fix (k

^{∗}_{1}

*, k*

_{2}

*) such that*

^{∗}*ϵ*

_{k}*∗*

1 *∈* Argmax(Q^{+}_{ii}*, x),* *ϵ*_{k}*∗*

2 *∈* Argmax(Q^{+}_{jj}*, x). For every* *ϵ*_{l}*∈*
Argmax(Q_{ij}*, x),*(*{k*^{∗}_{1}*, k*_{2}^{∗}*}, l)* is an edge in *G⃗*. Hence *η*_{k}*∗*

1 +*η*_{k}*∗*

2 *>*2η* _{l}*. Therefore

*Q*

^{(k}

_{ii}^{1}

^{∗}^{)}+

*Q*

^{(k}

_{jj}

^{∗}^{2}

^{)}+

*x*

^{(ρ)}

_{k}*∗*

1 +*x*^{(ρ)}_{k}*∗*

2 *>*2*|Q*^{(l)}_{ij}*|*+ 2x^{(ρ)}* _{l}* . As before, this implies that

*Q*

^{+}

*(x*

_{ii}^{(ρ)})

*⊙Q*

^{+}

*(x*

_{jj}^{(ρ)})

*>*(Q

*(x*

_{ij}^{(ρ)}))

^{⊙}^{2}for

*ρ >*0 small enough. Since we supposed that

*x*

*∈ S ∩*R

*, we have*

^{n}*x*

^{(ρ)}

*∈ T*for

*ρ*small enough.

76 Chapter 4. Tropical spectrahedra 1

2

0

Figure 4.3: The hypergraph from Example 4.38, consisting of the edges (0,1), (0,2), and
(*{*1,2*},*0).
particular, it is not regular. The hypergraph associated with (0,0,0) is depicted in Fig. 4.3.

This hypergraph admits a circulation (we put*γ**e*:= 1/3for all edges).

We now want to show that the condition of Lemma 4.37 is fulfilled generically.

**Lemma 4.39.** *There exists a set* *X⊂*T^{d}*with* *d*=*nm(m*+ 1)/2 *that fulfills the following two*
*conditions. First, every stratum of* *X* *is a finite union of hyperplanes. Second, ifQ*^{(1)}*, . . . , Q*^{(n)}
*is any sequence of symmetric tropical Metzler matrices such that the vector with entries* *|Q*^{(k)}_{ij}*|*
*(for* *i*⩽*j) does not belong to* *X, then the hypergraph* *G⃗**x* *does not admit a circulation for any*
*x∈*R^{n}*.*

where In_{1}(l) denotes the set of incoming edges with tails of cardinality 1 and In_{2}(l) denotes
the set of incoming edges with tails of cardinality 2. Since *γ* is a circulation, this expression
simplifies to

4.3. Genericity conditions 77
This set is a hyperplane. Indeed, suppose that the equality above is trivial (i.e., that it reduces
to 0 = 0). Take any edge *e* such that *γ*_{e}*̸*= 0 and any vertex *k* *∈* *T** _{e}*. Then, the coeﬃcient

*z*

^{(k)}

_{i}

_{e}

_{i}*appears on the left-hand side. Moreover, we have sign(Q*

_{e}^{(k)}

_{i}

_{e}

_{i}*) = 1. On the other hand, for every coeﬃcient*

_{e}*z*

^{(l)}

_{j}

_{e}

_{j}*that appears on the right-hand side we havesign(Q*

_{e}^{(l)}

_{j}

_{e}

_{j}*) =*

_{e}*−*1. This gives a contradiction.

Therefore, we can construct the stratum of *X* associated with *D* (denoted*X**D*) as follows:

we take all possible hypergraphs that can arise in our construction (since *n* is fixed, we have
finitely many of them). Out of them, we choose those hypergraphs that admit a circulation. For
every such hypergraph we pick exactly one circulation*γ*. After that, for every possible choice
of functions*e→i**e*,*e→*(i*e**, j**e*),^{1} we take a set*H*defined as in (4.8). If *H*is equal toR^{d}* ^{′}*, then
we ignore it. Otherwise,

*H*is a hyperplane. We take

*X*

*to be the union of all hyperplanes obtained in this way.*

_{D}The proof of Lemma 4.39 can be easily adapted to give a genericity condition both for Metzler and non-Metzler spectrahedra.

**Theorem 4.40.** *There exists a set* *X⊂*T^{d}*withd*=*nm(m*+ 1)/2 *that fulfills the following two*
*conditions. First, every stratum of* *X* *is a finite union of hyperplanes. Second, ifQ*^{(1)}*, . . . , Q*^{(n)}
*is any sequence of symmetric tropical matrices such that the vector with entries|Q*^{(k)}_{ij}*|(fori*⩽*j)*
*does not belong to* *X, then the matrices* *Q*^{(1)}*, . . . , Q*^{(n)} *fulfill Assumption A and for all* (Σ,♢),
this graph has vertices enumerated by numbers from *K). Suppose that this graph admits a*
circulation*γ*. As previously, for every edge *e*= (*{k*_{1}*, k*_{2}*}, l)* of *G⃗* we can take(i_{e}*, j** _{e}*)

*∈Σ*such

As previously, the set of all *z* *∈* R^{d}* ^{′}* that fulfills this equality is a hyperplane. Indeed, any
coeﬃcient

*z*

_{i}^{(k)}

*e**j**e* which appears on the left-hand side does not appear on the right-hand side
(note that here we use the fact that *Σ∩Σ*^{∁}=*∅). As before, we take all possible hypergraphs*

1Note that the dependence on*D*lies here, as the choice of*D*restricts the amount of possible functions*e**→**i**e*

*e**→*(i*e**, j**e*).

78 Chapter 4. Tropical spectrahedra
(where “all possible” takes into account the fact that *K* can vary), one circulation for each
hypergraph, all possible functions *e* *→* (i_{e}*, j** _{e}*) (the amount of such functions depends on

*D),*and all hyperplanes that can arise in this way. The union of these hyperplanes constitutes

*X*

*. We deduce the result from Lemma 4.37.*

_{D}