Genericity conditions

In this section we show that the requirements of Theorems 4.19 and 4.28 on the matricesQ^(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 pairG⃗:= (V, E), whereV is a finite set ofvertices andE is a finite set of(hyper)edges. Every edgee∈E is a pair(T_e, h_e), whereh_e∈V is called thehead of the edge, andT_e is a multiset with elements taken fromV. We call T_e themultiset of tails ofe. By|Te|we denote the cardinality ofTe(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_e. If 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 thathe=v. ByOut(v)we denote the multiset ofoutgoing edges, i.e., a multiset of edgesesuch thatv ∈T_e. We treatOut(v) as a multiset, with the convention thate∈E appears p times in Out(v) ifv appearsp 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∈E such thatγ_e ⩾0 for all e∈E,^∑_e∈Eγe = 1, and for all v∈V we have the equality

∑

e∈In(v)

|Te|γe= ^∑

e∈Out(v)

γe.

Observe that if a hypergraphG⃗ is fixed, then the set of all normalized circulations onG⃗ forms a polyhedron. We say that a hypergraphdoes 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 thekth vector of the standard basis in Rⁿ. Definition 4.35. Given a sequence of tropical symmetric Metzler matrices Q⁽¹⁾, . . . , Q⁽ⁿ⁾ ∈ T^m×m_± and a point x ∈ Rⁿ, we construct a hypergraph associated with x, denoted G⃗x, as follows:

• we putV := [n];

• for every i ∈[m] verifying Q⁺_ii(x) =Q⁻_ii(x) ̸=−∞, and every pair ϵ_k ∈ Argmax(Q⁺_ii, x), ϵ_l∈Argmax(Q⁻_ii, x), the hypergraphG⃗x contains an edge(k, l);

• for every i < j such that Q⁺_ii(x)⊙Q⁺_jj(x) = (Q_ij(x))^⊙2 ̸= −∞ and every triple ϵ_k1 ∈ Argmax(Q⁺_ii, x), ϵ_k2 ∈Argmax(Q⁺_jj, x), ϵ_l ∈Argmax(Q_ij, x), the hypergraph G⃗x contains an edge({k1, k2}, 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∈Rⁿ_⩾0:∀i∈ [p],⟨a_i, x⟩=b_i} is empty if and only if there exists a vectory∈R^p such that^∑p_i=1b_iy_i <0 and

∑p

i=1a_iky_i ⩾0 for all k∈[n].

Lemma 4.37. Suppose that for every x∈Rⁿ the hypergraphG⃗x does not admit a circulation.

Then, the matrices Q⁽¹⁾, . . . , Q⁽ⁿ⁾ fulfill Assumption A and S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾)∩Rⁿ is regular.

Proof. To prove the first part, suppose that we have Q^(k)_ii +Q^(k)_jj = 2|Q^(k)_ij | for somei̸=j and Q^(k)_ii , Q^(k)_jj ∈T+. Take the pointx:=N ϵk∈Rⁿ. IfN is large enough, then we have

Q⁺_ii(x)⊙Q⁺_jj(x) =Q^(k)_ii +Q^(k)_jj + 2N = 2|Q^(k)_ij |+ 2N = (Q_ij(x))^⊙2

and the hypergraphG⃗x contains the edge({k, k}, k). This hypergraph admits a circulation (we putγe := 1fore= ({k, k}, k) and γe:= 0 for other edges), which gives a contradiction.

We now claim that the set S ∩Rⁿ is regular. Let T be defined as in Definition 4.16. Let us show that for every x ∈ S ∩Rⁿ there exists a vector η ∈ Rⁿ such that x+ρη belongs to T for ρ >0 small enough. This is sufficient to prove the claim because T ∩Rⁿ is a subset of the interior of S ∩Rⁿ (as shown in the proof of Theorem 4.19). Fix a point x ∈ S ∩Rⁿ. If x belongs to T, then we can take η := 0. Otherwise, let G⃗x denote the hypergraph associated withx. The polytope of circulations of this hypergraph is empty. Therefore, by Farkas’ lemma (Theorem 4.36), there exists a vector η∈Rⁿ such that for every edge e∈E we have

∑

v∈Te

η_v >|T_e|η_he.

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

First, suppose that there is i∈[m]such thatQ⁺_ii(x) =Q⁻_ii(x)̸=−∞. Fix anyk^∗ such that ϵk^∗ ∈Argmax(Q⁺_ii, x) and take any l such that ϵl ∈Argmax(Q⁻_ii, x). Then (k^∗, l) is an edge in G⃗x. Thereforeη_k∗ > η_l. Moreover,Q^(k_ii^∗)+x_k∗ =|Q^(l)_ii |+x_land henceQ^(k_ii^∗)+x^(ρ)_k∗ >|Q^(l)_ii|+x^(ρ)_l . Furthermore, for every 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′ forρ 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⁻_ii(x^(ρ)).

The second case is analogous. If there isi < jsuch thatQ⁺_ii(x)⊙Q⁺_jj(x) = (Qij(x))^⊙2 ̸=−∞, then we fix (k^∗₁, k₂^∗) 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, k₂^∗}, l) is an edge in G⃗. Hence η_k∗

1 +η_k∗

2 >2η_l. ThereforeQ^(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⁺_ii(x^(ρ))⊙Q⁺_jj(x^(ρ))>(Q_ij(x^(ρ)))^⊙2 for ρ > 0 small enough. Since we supposed that x ∈ S ∩Rⁿ, we have 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 putγe:= 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 G⃗x does not admit a circulation for any x∈Rⁿ.

where In₁(l) denotes the set of incoming edges with tails of cardinality 1 and In₂(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)_ieie appears on the left-hand side. Moreover, we have sign(Q^(k)_ieie) = 1. On the other hand, for every coefficientz^(l)_jeje that appears on the right-hand side we havesign(Q^(l)_jeje) =−1. This gives a contradiction.

Therefore, we can construct the stratum of X associated with D (denotedXD) 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 circulationγ. After that, for every possible choice of functionse→ie,e→(ie, je),¹ we take a setHdefined as in (4.8). If His equal toR^d′, then we ignore it. Otherwise, H is a hyperplane. We take X_D to be the union of all hyperplanes obtained in this way.

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^dwithd=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 circulationγ. As previously, for every edge e= ({k₁, k₂}, 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)

eje 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 onDlies here, as the choice ofDrestricts the amount of possible functionse→ie

e→(ie, je).

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 constitutesX_D. We deduce the result from Lemma 4.37.