Tropical polynomials

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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 X1, . . . , Xn is a formal expression of the form

P(X) =

αΛ

aα⊙X1α1⊙ · · · ⊙Xnαn, (2.6) whereΛis a finite subset of{0,1,2, . . .}n, andaα T±\{−∞}for allα∈Λ. Since the addition inT± is only partially defined, we cannot evaluate P(x) for all the points x∈Tn±. However, if x∈Tn± is such that all of the termsaα⊙x1α1 ⊙ · · · ⊙xnαn have the same sign, then we define P(x) as the tropical sum of these terms.

Example2.45. IfP(X) = 2⊙X13⊙X24(0⊙X2), thenP(1,5) = 25,P(⊖1,5) =(5), whereasP is not defined for (1,−5/3).

The next definition and lemma shows that a nonzero tropical polynomial divides Rn into cells that form a polyhedral complex. The union of the boundaries of cells of this complex is called atropical hypersurface—we refer to [MS15, Section 3.1] for more information.

Definition 2.46. Given a tropical polynomialP as in (2.6), we say thatP isnonzero if the set Λ is nonempty. For every such tropical polynomial and every pointx∈Rnwe define theset of maximizing multi-indices at x as

Argmax(P, x) :={α∈Λ:∀β ∈Λ, |aα|+⟨α, x⟩|aβ|+⟨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

Wα := clR({xRn: 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

WL:= clR({xRn: Argmax(P, x) =L}) (2.7) is a (possibly empty) face of Wα. Moreover, the family{WL}LΛ is a polyhedral complex whose support is equal to Rn. (Here, clR(·) refers to the closure in Rn and we use the convention that W =∅.)

Sketch of the proof. To prove the first statement, note that a pointx∈Rnsatisfies the equality Argmax(P, x) = α if and only if the inequality |aα|+⟨α, x⟩ > |aβ|+⟨β, x⟩ is true for all β ∈Λ\ {α}. Therefore, if Wα is nonempty, then it contains a ball. A relative interior of a face ofWα is obtained by fixing a subset of inequalities of the form|aα|+⟨α, x⟩>|aβ|+⟨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 Wα if and only if V = clR({x Rn: Argmax(P, x) =L}) for some L ⊂Λ such thatα ∈L. The polyhedra WL

have disjoint relative interiors, and hence the family {WL}LΛ is a polyhedral complex. The fact that its support is equal to Rn follows from the definition.

Example 2.48. Take

P(X1, X2) :=(2⊙X1)((4)⊙X12))((3)⊙X1⊙X2)(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 ofP).

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 polynomialP as in (2.6), we set Λ+ := ∈Λ:aα T+} and Λ := ∈Λ: aα T}. Furthermore, we define P+ :=α∈Λ+aα⊙X1α1 ⊙ · · · ⊙Xnαn and P:=αΛ|aα| ⊙X1α1 ⊙ · · · ⊙Xnαn. Finally, if P is nonzero, then we set

S(P) :={x∈Rn:P+(x)⩾P(x)}

(with the convention thatP+(x) :=−∞ifΛ+= and P(x) :=−∞ ifΛ=∅).

Remark 2.50. We point out the valuesP+(x), P(x) are well defined for allx∈Tn. Therefore, S(P) is also well defined.

The following definition and a lemma show the connection between polynomial inequalities overK0 and tropical polynomial inequalities.

Definition 2.51. To any polynomial P(X) =

αΛ

aαX1α1. . . Xnαn K[X1, . . . , Xn] (2.8) over Puiseux series we associate a tropical polynomial defined as

trop(P) :=

αΛ

sval(aα)⊙X1α1⊙ · · · ⊙Xnαn.

28 Chapter 2. Preliminaries

Figure 2.2: Support of the polyhedral complexC(P).

Lemma 2.52. Let P K[X1, . . . , Xn] be a polynomial and let P := trop(P). If x Kn0 is such that P(x)⩾0, then P+(val(x))P(val(x)).

Proof. Define x=val(x)Tn± and let

P+(X) =

αΛ+

aαX1α1. . . Xnαn P(X) =

αΛ

|aα|X1α1. . . Xnαn.

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 WL ∈ C(P) as in (2.7) is positive if there exists at least oneα∈Lsuch thataα T+ or ifWL 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 toS(P).

Sketch of the proof. As in the proof of Lemma 2.47, if WL belongs to C(P) and V is its face, then V =WL for someL ⊂L. This means that all faces of WL 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 thatx∈S(P) if and only if Argmax(P, x)∩Λ+̸=.

Example 2.55. Let P(X1, X2) 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 P1, . . . , Pm, we also regard the refinements of complexes defined by P1, . . . , Pm. More precisely, we make the following definition.

Definition 2.56. IfP1, . . . , Pm are nonzero tropical polynomials, then we defineC(P1, . . . , Pm) and C(P1, . . . , Pm) as

C(P1, . . . , Pm) :={mi=1Vi:∀i, Vi∈ C(Pi)}, C(P1, . . . , Pm) :={mi=1Vi:∀i, Vi∈ C(Pi)}.

2.4. Tropical polynomials 29 Lemma 2.57. The families C(P1, . . . , Pm) and C(P1, . . . , Pm) are polyhedral complexes. The support of the former is equal to Rn while the support of the latter coincides with

S(P1, . . . , Pm) :={x∈Rn:∀i, Pi+(x)⩾Pi(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 Rn is called regular if X = clR(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 cellW of C, then x ∈ W = clR(int(W)) clR(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 ballB(y, r)Rn 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. Ify 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 thatr < r. Hence,y belongs to some full-dimensional cell ofC.

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. SinceW is closed, we get x∈ W.

Lemma 2.59. Suppose that the polyhedral complexC(P1, . . . , Pm)has a regular support. Then this support, S(P1, . . . , Pm), coincides with the closure of the set

S>(P1, . . . , Pm) :={x∈Rn:∀i, Pi+(x)> Pi(x)}.

Proof. If there is a tropical polynomialPi such thatP =Pi, then both of these sets are empty and the claim is trivially true. If there is a tropical polynomialPisuch thatP =Pi+, then we can remove this polynomial from the collection and this does not change the sets S(P1, . . . , Pm) and S>(P1, . . . , Pm). Hence, we can suppose that all the tropical polynomials Pi+, Pi are nonzero. Then, the setS(P1, . . . , Pm)is closed since the tropical polynomial functionsPi±are continuous. We obviously haveS>(P1, . . . , Pm)S(P1, . . . , Pm). Therefore,

clR(S>(P1, . . . , Pm))S(P1, . . . , Pm).

Consider nowy∈S(P1, . . . , Pm). Since this set is regular, by Lemma 2.58,ybelongs to a full-dimensional cellW ofC(P1, . . . , Pm). We haveW =1imWi, whereWiis a full-dimensional cell of C(Pi). This implies that Wi = clR({x Rn: Argmax(Pi, x) = Li}) where Li is a one element subset of Λ+(Pi). We conclude that Pi+(x) > Pi(x) holds for all x int(Wi), and so, int(W) S>(P1, . . . , Pm). We have W = clR(int(W)) by Lemma 2.3 and hence y∈clR(S>(P1, . . . , Pm)).

30 Chapter 2. Preliminaries

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