Tropical Helton–Nie conjecture for Puiseux series

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We perform the first transformation on the graph G⃗ and denote the transformed graph by G⃗1. We then perform the second transformation on every edge inG⃗1that joins two Random vertices.

We denote the transformed graph byG⃗ and the associated operator asF:Rn+n Rn+n. By Lemmas 5.42 and 5.43, the real tropical cone {x∈Rn:xF(x)} is the projection of the real tropical cone{(x, x)Rn×Rn: (x, x)⩽F(x, x)}. Furthermore,G⃗ fulfills the conditions of Proposition 5.40. Therefore, the set S ={(x, x)Tn×Tn: (x, x) ⩽F(x, x)} is a tropical Metzler spectrahedral cone. Finally, we take the set

S′′={(x, x, y)∈Tn×Tn×Tn+n: (x, x)⩽F(x, x)(x, x) +y⩾0}

={(x, x, y)∈Rn×Rn×Rn+n: (x, x)⩽F(x, x)(x, x) +y⩾0}.

The setS′′ is a real tropical Metzler spectrahedron because a constraint of the formxv+yv⩾0 can be encoded by adding a2×2block

[xv 0

⊖0 yv

]

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 existsy Rn+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 inTn. 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.,SRn), then we define itsreal homogenization as

Srh :={(x0, x0+x)∈ Rn+1:x∈S}. IfSTn is a tropically convex set, then we define itshomogenization as

Sh:={(x0, x0+x)∈ Tn+1:x∈S}.

If S(Q(0)|Q(1), . . . , Q(n)) Tn is a tropical Metzler spectrahedron, then we define its formal homogenization Sf hTn+1 as the tropical Metzler spectrahedral cone

Sf h:=S(Q(0), Q(1), . . . , Q(n)).

Lemma 5.47. If S Tn is tropically convex, then its homogenization is a tropical cone.

Moreover, if Sis included in Rn, 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 Tn is tropically convex, take its homogenization Sh, two points (x0, x0+x),(y0, y0+y)∈Srh (such thatx, y∈Sand x0, y0 T) andλ, 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 Srh = {(x, y) R2:yx}, while its homogenization is equal to Sh = Srh∪ {−∞}. More-over, S=S(Q(0)|Q(1)) is a tropical Metzler spectrahedron defined by the matricesQ(0) =⊖0, Q(1)= 0. Then, its formal homogenization is equal toSf h={(x, y)T2:yx}.

Lemma 5.49. Every closed, semilinear, tropically convex set in Rn is a projection of a real tropical Metzler spectrahedron.

Proof. Take any closed, semilinear, tropically convex set SRn and consider its real homog-enizationSrh. This set is a real tropical cone in Rn+1 by Lemma 5.47. Moreover, it is closed

5.5. Tropical Helton–Nie conjecture for Puiseux series 101 and definable in Log. This implies that it is semilinear by Lemma 2.103. By Proposition 5.44, Srh is a projection of a real tropical Metzler spectrahedronS1 R×Rn×Rn. Consider the set

S2={(x0, x, y)∈R×Rn×Rn: (x0, x, y)∈ S1∧x0= 0}.

The set S2 is a tropical Metzler spectrahedron because the constraint x0 = 0 can be encoded by adding a2×2 block [

0⊕x0 −∞

−∞ 0(⊖x0) ]

to the matrices that describe S1. Furthermore, S is its projection. Indeed, if x S, then (0, x)Srh and there existsy Rn such that(0, x, y) ∈ S1. Conversely, if(0, x, y)∈ S1, then (0, x)Srh and hencex∈S.

We now want to extend the result of Lemma 5.49 to tropically convex sets in Tn. 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 STn is a projected tropical Metzler spectrahedron if and only if its homogenization is a projected tropical Metzler spectrahedron.

Proof. First, suppose thatSh is a projection of a tropical Metzler spectrahedronS1T×Tn× Tn. Consider the set

S2 ={(x0, x, y)∈T×Tn×Tn: (x0, x, y)∈ S1∧x0 = 0}.

As in the proof of Lemma 5.49, the set S2 is a tropical Metzler spectrahedron and S is its projection. Conversely, suppose that S is a projection of a tropical Metzler spectrahedron S1 Tn×Tn. Consider its formal homogenizationS1f hT1+n+n and take the set

S2={(x0, x, y, z)T×Tn×Tn×Tn: (x0, x, y)∈ S1f h ∧ ∀k∈[n], x0+zk⩾2xk}. The set S2 is a tropical Metzler spectrahedron because a constraint of the form x0+zk ⩾2xk

can be encoded by adding a2×2block

[ x0 ⊖xk

⊖xk zk

]

to the matrices that describe S1. We will show that Sh is a projection of S2. To see this, take any point (x0, x0 +x) Sh, where x S and x0 T. If x0 = −∞, then −∞ ∈ Sh belongs to the projection of S2. Otherwise, take y such that (x, y) ∈ S1 and observe that we have (x0, x0 +x, x0 +y) ∈ S1f h. Since x0 R, if we take zk large enough, then (x0, x0+x, x0+y, z)∈ S2. This shows thatSh is included in the projection of S2. Conversely, suppose that (x0, x, y, z) ∈ S2. If x0 = −∞, then x = −∞ and hence (x0, x) Sh. If x0 ̸=−∞, then we have(0,−x0+x,−x0+y,−x0+z)∈ S2. Hence(0,−x0+x,−x0+y)∈ S1f h, (−x0+x,−x0+y)∈ S1, and −x0+x∈S. Therefore(x0, x)∈Sh.

102 Chapter 5. Tropical analogue of the Helton–Nie conjecture Lemma 5.51. Suppose that S1,S2 Tn are projected tropical Metzler spectrahedra. Then tconv(S1S2) is a projected tropical Metzler spectrahedron.

Proof. LetS= tconv(S1S2)and consider

S1hS2h :={x∈Tn+1:∃(u, w)∈S1h×S2h, x=u⊕w}.

First, we will show that we have the identity Sh = S1hS2h. Indeed, since S1 S, we have S1h Sh. Similarly, S2h Sh. Therefore, we have S1hS2h Sh. Conversely, take a point z∈Sh. By Lemma 5.17, we can write z as

z= (

z0, z0(⊙x)⊕⊙y)))Sh, whereλ⊕µ= 0,x∈S1, and y∈S2. Then z= ˜x⊕y, where˜

˜

x:= (λ⊙z0,⊙z0)⊙x)∈S1h,

˜

y:= (µ⊙z0,⊙z0)⊙y)∈S2h.

HenceSh =S1hS2h. SinceS1,S2 are projected tropical Metzler spectrahedra, the same is true for S1h,S2h by Lemma 5.50. Let S1h be a projection of S1 Tn+1+n1 and S2h be a projection of S2 Tn+1+n2. Then, the set S1h×S2h is a projection of the set S1× S2. Moreover, the setS1× S2 is a tropical Metzler spectrahedron (described by block-diagonal matrices such that the first block is given by the matrices that describe S1 and the second block is given by the matrices that describeS2). Therefore, the set

S3:={(u, u, w, w, x) : T(n+1)+n1+(n+1)+n2+(n+1): (u, u, w, w)∈ S1× S2, x=u⊕w} is a tropical Metzler spectrahedron (because a constraint of the form xk = uk⊕wk can be encoded by adding a2×2block to the matrices that describeS1× S2). Moreover,S1hS2h is a projection ofS3. Since Sh=S1hS2h, the set Sis a projected tropical Metzler spectrahedron by Lemma 5.50.

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 Tn 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 stratumSK RK is nonempty. The setSK is a projection of a real tropical Metzler spectrahedron SK RK×Rn. For any x∈Tn we denote by xK TK the subvector formed by the coordinates of xwith indices inK. Furthermore, let XK Tn denote the set

XK :={x∈Tn:xk̸=−∞ ⇐⇒ k∈K}.

The setSXK is a projection of a tropical Metzler spectrahedron defined as S˜K ={(x, y)Tn×Tn: (xK, y)∈ SK ∧ ∀k /∈K,−∞xk}.

Moreover, forK=, let us denoteX =−∞. Note that the intersectionSXis either empty or is equal to−∞, and that−∞is a tropical Metzler spectrahedron (defined by the inequalities

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