the set S⊂Kn+m+1 defined as is an injective semialgebraic function from Km to A. Hence, by Proposition 2.20, we have dim(A) =m. Furthermore, note that f(]0,1/m[m) is a subset of∆. Sincef is injective, using Proposition 2.20 again we havem= dim(f(]0,1/m[m))⩽dim(∆)⩽dim(A) =m.
Corollary 2.26. Suppose that the set S⊂Kn is nonempty, semialgebraic, and convex. Then, the dimension of S is equal to the largest dimension of a simplex contained in S. Equivalently, is it equal to the smallest dimension of an affine space that contains S.
Sketch of the proof. Take the largest collection u(1), . . . , u(m) ∈ Kn of linearly independent vectors such that there existsu(0)∈Knsatisfying∆(u(0), u(1), . . . , u(m))⊂S (we allowm= 0).
LetAdenote the affine space u(0)+ span(u(1), . . . , u(m)). If there exists a pointx∈S\A, then the vectors u(1), . . . , u(m), x−u(0) ∈ Kn are linearly independent. By convexity of S, this implies that the simplex ∆(u(0), u(1), . . . , u(m), x−u(0)) ∈Kn is contained inS, which gives a contradiction. Hence S ⊂A. Therefore, by Lemma 2.25 we have dim(S) =m. Moreover, m is the smallest dimension of an affine space that contains S.
2.3 Puiseux series and tropical semifield
In this section, we discuss the fundamental objects of this thesis—the field of Puiseux series and the tropical semifield. We start by briefly introducing the field of Puiseux series. More information about this field can be found in Appendix A, where we discuss in detail the dif-ferent fields used in tropical geometry (e.g., Puiseux series with rational exponents, generalized Puiseux series, Hahn series, both formal and convergent), prove their basic properties, and, following van den Dries and Speissegger [vdDS98], prove that all these fields are real closed (or algebraically closed in the case of complex coefficients). In this work, we use the field of con-vergent, generalized, real Puiseux series. In the context of tropical geometry, a variant of this field without without the convergence assumption first appeared in [Mar10]. Up to a change of variables, this field is isomorphic to the field of generalized Dirichlet series studied by Hardy and Riesz [HR15].
24 Chapter 2. Preliminaries Definition 2.27. Anabsolutely convergent generalized real Puiseux series is a series of the form
x=
∑∞ i=1
cλitλi, (2.2)
where tis a formal parameter, (λi)i⩾1 is a strictly decreasing sequence of real numbers that is either finite or unbounded, and cλi ∈ R\ {0}. Furthermore, the series (2.2) is required to be absolutely convergent fortlarge enough. There is also a special, empty series, which is denoted by 0.
Remark 2.28. For simplicity, throughout this work we refer to absolutely convergent generalized real Puiseux series as “Puiseux series.” We denote the set of Puiseux series byK.
Definition 2.29. We denote by lc(x) the coefficient cλ1 of the leading term in the series xas in (2.2), with the convention that lc(0) = 0. We also endowK with a linear order ⩽, which is defined asy⩽xiflc(x−y)⩾0. Equivalently, we havey⩽xif and only ify(t)⩽x(t) for all sufficiently large t >0. We denote byK⩾0 the set of nonnegative seriesx, i.e., the set of series satisfying x⩾0.
The Puiseux series can be added and multiplied in the natural way. Furthermore, given the order introduced in Definition 2.29, the ring of Puiseux series forms a real closed field. This was proven by van den Dries and Speissegger [vdDS98].
Theorem 2.30 ([vdDS98]). The set of Puiseux seriesKendowed with the order⩽forms a real closed field.
The tropical semifield describes the algebraic structure ofK under the valuation map.
Definition 2.31. The valuation of an element x ∈ K as in (2.2) is defined as the greatest exponentλ1 occurring in the series,val(x) :=λ1. Equivalently, the valuation is given by
val(x) = lim
t→+∞logt|x(t)|,
wherelogt(z) := log(z)/log(t). We use the convention that val(0) =−∞.
Definition 2.32. Thetropical semifield is a structureT= (T,⊕,⊙), where the underlying set T is defined as T:=R∪ {−∞}, the tropical addition is defined as x⊕y = max(x, y), and the tropical multiplication is defined as x⊙y=x+y.
Remark 2.33. We point out that −∞ is the neutral element of the tropical addition, and 0 is the neutral element of the tropical multiplication. The word “semifield” refers to the fact that the tropical addition does not have an inverse (but all other properties of fields are satisfied by T). Throughout this work, we use the notation⊕ni=1ai =a1⊕ · · · ⊕an and a⊙n =a⊙ · · · ⊙a (n times).
Remark 2.34. We endow T with the standard order ⩽. Since T is totally ordered, it has a topology defined by the order, i.e., the smallest topology such that all sets of the form]a, b[and [−∞, b[fora, b∈Rare open. We extend this topology to Tn by taking the product topology.
In this work, we only use a few basic properties of the tropical semifield. We refer to [But10]
for more information about the algebraic properties ofT.The following observation relates the valuation overK to the tropical semifieldT.
2.3. Puiseux series and tropical semifield 25 Lemma 2.35. The valuation val:K→T satisfies the properties
val(x+y)⩽max(val(x),val(y)) (2.3)
val(xy) =val(x) +val(y). (2.4)
Furthermore, for every x⩾y⩾0 we have
val(x)⩾val(y). (2.5)
Proof. The proof of the first two properties follows from the definition of addition and multi-plication of the series. More precisely, the greatest exponent of the sum of two series cannot be bigger that the greatest exponent of each of them, and the greatest exponent of the product is equal to the sum of the greatest exponents of the factors. To prove the third property, note that if val(y)>val(x), then lc(x−y) =−lc(y)<0, which gives a contradiction with the fact thatx⩾y.
Remark 2.36. We point out that the equality holds in (2.3) if the leading terms ofxand y do not cancel. This is the case, for instance, ifval(x)̸=val(y)or if x,y⩾0.
Remark 2.37. The properties given in Lemma 2.35 imply that the valuation is an order-preserving morphism of semifields fromK⩾0 toT.
When dealing with semialgebraic sets, it is convenient to keep track not only of the valuations of the elements ofK, but also of their signs. To this end, we introduce the set of signed tropical numbers and signed valuation.
Definition 2.38. The set of signed tropical numbers is defined as T± := ({+1,−1} ×R)∪ {(0,−∞)}. The modulus function |·|: T± → T is defined as the projection which forgets the first coordinate. The elements of the form (1, a) ofT± are called positive tropical numbers and are denoted byT+. Similarly, the elements of the form(−1, a)ofT±are callednegative tropical numbers and are denoted by T−. By convention, we denote the positive tropical number(1, a) bya, the negative tropical number(−1, a) by⊖a, and the element(0,−∞) by−∞. Here, ⊖is a formal symbol.
Definition 2.39. We extend the definition of tropical multiplication toT±using the usual rules for signs. In other words, for δ1, δ2 ∈ {−1,0,+1} and a, b ∈ T such that (δ1, a),(δ2, b) ∈ T± we define (δ1, a)⊙(δ2, b) := (δ1δ2, a+b). We also partially extend the tropical addition to the elements of T± which have the same sign. In other words, if δ1 = δ2, then we define (δ1, a)⊕(δ1, b) := (δ1,max{a, b}).
Example 2.40. We have(⊖3)⊙7 =⊖10,(⊖3)⊙(⊖7) = 10, 3⊕7 = 7, and (⊖3)⊕(⊖7) =⊖7, but(⊖3)⊕7is not defined.
Let us note that one can extend the set T± further in oder to get a semiring with a well-defined tropical addition [AGG09], or work with hyperfields [Vir10, CC11, BB16] instead of semifields to have a well-defined, but multivalued addition. However, the partial addition defined above is sufficient for this work.
Remark 2.41. We point out that the tropical semiring Tis isomorphic toT+∪ {−∞}.
Definition 2.42. We define the sign function sign:K→ {−1,0,+1} assign(x) = 1 if x>0, sign(x) = −1 if x <0, and sign(0) = 0. We define the absolute value function |·|:K → K⩾0
as |x| = sign(x)x. Furthermore, we define the signed valuation sval:K → T± as sval(x) = (sign(x),val(x)).
26 Chapter 2. Preliminaries Remark 2.43. We extend the functions val:K→T,sval:K→ T±, and |·|:T± →T to vectors and matrices by applying them coordinatewise.