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A brief introduction to real dessins d’enfant
For more details, see [Ore03, Bru06, Bih07] for example. Consider a real rational map ϕ = QP : C → C, where P and Q are two real polynomials. The degree of ϕ is the maximum of the degrees of P and Q. We extend ϕ to a rational homogeneous function CP 1 → CP 1, (x0 : x1) (1 : P/Q), that we denote again by ϕ. Define := ϕ−1(RP 1).
This is a real graph on CP 1 invariant with respect to the complex conjugation and which contains RP 1. Any connected component of CP 1 \ is homeomorphic to an open disk. Moreover, each vertex of has even valency, and the multiplicity of a critical point with real critical value of ϕ is half its valency. The graph contains the inverse images of (1 : 0), (0 : 1) and (1 : 1), which are the sets of roots of P , Q and P/Q − 1 respectively. Denote by the same letter p (resp. q and r) the points of which are mapped to (1 : 0) (resp. (0 : 1) and (1 : 1)). Orient the real axis on the target space via the arrows 0 → ∞ → 1 → 0 (orientation given by the decreasing order in R), which is equivalent to orienting RP 1 via the arrows (1 : 0) → (0 : 1) → (1 : 1). Pull back this orientation by ϕ, the graph becomes an oriented graph, with the orientation given by arrows p → q → r → p. A cycle of is the boundary of a connected component of CP 1\ . Any such cycle contains the same non-zero number of letters r, p , q (see Figure 2.1). We say that a cycle obeys the cycle rule. The graph is called real dessin d’enfant associated to ϕ. Since is invariant under complex conjugation, it is determined by its intersection with one connected component H (for half) of CP 1 \ RP 1. Since ϕ is real, its degree is the sum of the degrees of its restrictions to connected components of CP 1 \ . To represent the real dessin d’enfant, we draw a horizontal line corresponding to the real projective line and draw below one half H of , see Figure 3.1 for instance.
Clearly, the arrangement of real roots of P , Q and P/Q − 1 together with their multiplicities can be extracted from the graph . We encode this arrangement together with the multiplicities by what is called a root scheme.
Definition 2.1 ([Bru06, Ore03]). A root scheme is a k-tuple (l1, m1), . . . , (lk, mk) ∈ ({p, q, r} × N)k. A root scheme is realizable by polynomials of degree d if there exist real polynomials P and Q such that ϕ has degree d and if x1 < . . . < xk are the real roots of P , Q and P/Q − 1, then li = p (resp. q, r) if xi is a root of P (resp. Q, P/Q − 1) and mi is the multiplicity of xi.
Conversely, suppose we are given a real graph ⊂ CP 1 that is invariant under complex conjugation, together with a real continuous map φ : → RP 1. Denote the inverse images of 0, ∞ and 1 by letters p, q and r, respectively, and orient with the pull back by φ of the above orientation of RP 1. This graph is called a real rational graph [Bru06] if any vertex of has even valency and any connected component of CP 1 \ is homeomorphic to an open disk. Then, for any connected component D of CP 1 \ , the map φ|∂D is a covering of RP 1 whose degree dD is the number of letters p (resp. q, r) in ∂D. We define the degree of to be half the sum of the degrees dD over all connected components of CP 1 \ . Since φ is a real map, the degree of is also the sum of the degrees dD over all connected components D of CP 1 \ contained in one connected component of CP 1 \ RP 1.
The following result [Ore03] explains the importance of real rational graphs in computing the roots of P/Q − 1.
Proposition 2.2 (Orevkov). A root scheme is realizable by polynomials of degree d if and only if it can be extracted from a real rational graph of degree d on CP 1.
We show now how to prove the if part in Proposition 2.2 (see [Bih07, Bru06, Ore03]). For each connected component D of CP 1 \ , extend φ|∂D to a branched covering of degree dD (use the map
z zdD ) of one connected component of CP 1 \ RP 1, so that two adjacent connected components of CP 1 \ project to different connected components of CP 1 \ RP 1. Then, it is possible to glue continuously these maps in order to obtain a real branched covering φ : CP 1 → CP 1 of degree d. The map φ becomes a real rational map of degree d for the standard complex structure on the target space and its pull-back by φ on the source space. There exist then real polynomials P and Q such that P/Q has degree d and φ = P/Q, so that the points p (resp. q, r) correspond to the roots of P (resp. Q, P/Q − 1) and = φ−1(RP 1).
A brief introduction to tropical geometry
The notations in this section are taken from [BLdM12, BB13, Ren15, GL15].
Polytopes and subdivisions
Let Rn denote the n-dimensional Euclidean space, endowed with the standard inner product , : Rn × R → R.
Definition 2.3. A rational polyhedron in Rn is a convex set of points x, defined by a finite number of inequalities of type x, w ≤ c, where w ∈ Zn and c ∈ Rn.
If a rational polyhedron is closed, then it is called an integer convex polytope. All polytopes considered in Chapter 6 are integer convex.
Definition 2.4. A rational polyhedral complex is a finite set of rational polyhedra P = {Δi}i such that
1. for every Δ ∈ P, if Δ′ is a face of Δ, then Δ′ ∈ P, and
2. if Δ, Δ′ ∈ P, then Δ ∩ Δ′ is a face of both Δ and Δ′.
Let F be a field of characteristic zero. For z = (z1, . . . , zn) ∈ F n and w = (w1, . . . , wn) ∈ Rn, set zw = zw1 • • • zwn . Consider a polynomial f = cwzw ∈ F [z±1, . . . , z±1], with W = ∅ a 1 n w∈W 1 n finite subset of Zn, and cw ∈ F ∗.
Definition 2.5. The Newton polytope Δ(f ) of f is defined to be the convex hull Conv(W) of W .
Definition 2.6. A polyhedral subdivision of an integer convex polytope Δ is a set of integer convex polytopes {Δi}i∈I such that
• ∪i∈I Δi = Δ, and
• if i, j ∈ I, then if the intersection Δi ∩ Δj is non-empty, it is a common face of the polytope Δi and the polytope Δj .
Definition 2.7. Let Δ be an integer convex polytope in Rn and let τ denote a polyhedral subdivision of Δ consisting of integer convex polytopes. We say that τ is regular if there exists a continuous, convex, piecewise-linear function ϕ : Δ → R which is affine linear on every simplex of τ .
Let Δ be an integer convex polytope in Rn and let φ : Δ ∩ Zn → R be a function. We denote by Δ(φ) the convex hull of the graph of φ, i.e., ˆ {(i, φ(i)) ∈ R n+1 | i ∈ Δ ∩ Z n } . Δ(φ) := Conv
Then the polyhedral subdivision of Δ, induced by projecting the union of the lower faces of Δ(φ) onto the first n coordinates, is regular. In the following, we describe how we define φ using the polynomials that we will be working with.
Tropical polynomials and hypersurfaces
A locally convergent generalized Puiseux series is a formal series of the form a(t) = αrtr, r∈R where R ⊂ R is a well-ordered set, all αr ∈ C, and the series is convergent for t > 0 small enough. We denote by K the set of all locally convergent generalized Puiseux series. It is naturally a field of characteristic 0, which turns out to be algebraically closed.
Notation 2.8. Let coef(a(t)) denote the coefficient of the first term of a(t) following the increasing order of the exponents of t. We extend coef to a map Coef : Kn → Rn by taking coef coordinate-wise, i.e. Coef(a1(t), . . . , an(t)) = (coef(a1(t)), . . . , coef(an(t)))
An element a(t) = αrtr of K is said to be real if αr ∈ R for all r, and positive if a(t) is r∈R real and coef(a(t)) > 0.
Denote by RK (resp. RK>0) the subfield of K composed of real (resp. positive) series. Since elements of K are convergent for t > 0 small enough, an algebraic variety over K (resp. RK) can be seen as a one parametric family of algebraic varieties over C (resp. R). The field K has a natural non-archimedian valuation defined as follows:
val : K −→ R ∪ {−∞}
0 → −∞
αrtr = 0 → − minR{r | αr = 0}.
r∈R
The valuation extends naturally to a map Val : Kn → (R ∪ {−∞})n by evaluating val coordinate-wise, i.e. Val(z1, . . . , zn) = (val(z1), . . . , val(zn)). We shall often use the notation val and Val when the context is a tropical polynomial or a tropical hypersurface. On the other hand, define ord := − val, with ord(0) = +∞, and use it as a notation when the context is an element in RKn or a polynomial in RK[z1±1, . . . , z2±1].
Convention 2.9. For any s ∈ K, we have coef(s) = 0 ⇔ s = 0 and ord(s) = +∞ ⇔ s = 0
Consider a polynomial f (z) := cwzw ∈ K[z1±1, . . . , zn±1], with W a finite subset of Zn and all cw set of f in (K∗)n
The tropical hypersurface Vftrop image under Val of Vf : are non-zero. Let Vf = {z ∈ (K∗)2 | f (z) = 0} be the zero associated to f is the closure (in the usual topology) of the Vftrop = Val(Vf ) ⊂ Rn, endowed with a weight function which we will define later. There are other equivalent definitions of a tropical hypersurface. Namely, define
ν: W −→ R
w → ord(cw).
Its Legendre transform is a piecewise-linear convex function
L(ν) : Rn −→ R
x → max{ x, w − ν(w)}.
w W
We have the fundamental Theorem of Kapranov [Kap00].
Theorem 2.10 (Kapranov). A tropical hypersurface Vftrop is the corner locus of L(ν).
The corner locus of L(ν) is the set of points at which it is not differentiable. Tropical hypersur-faces can also be described as algebraic varieties over the tropical semifield (R∪ {−∞}, “ + ”, “×”), where for any two elements x and y in R ∪ {−∞}, one has “x + y” = max(x, y) and “x × y” = x + y.
A multivariate tropical polynomial is a polynomial in R[x1, . . . , xn], where the addition and multi-plication are the tropical ones. Hence, a tropical polynomial is given by a maximum of finitely many affine functions whose linear parts have integer coefficients and constant parts are real numbers. The tropicalization of the polynomial f is a tropical polynomial ftrop(x) = max{ x, w + val(cw)}.
This tropical polynomial coincides with the piecewise-linear convex function L(ν) defined above. Therefore, Theorem 2.10 asserts that Vftrop is the corner locus of ftrop. Conversely, the corner locus of any tropical polynomial is a tropical hypersurface.
Tropical hypersurfaces and subdivisions
A tropical hypersurface induces a subdivision of the Newton polytope Δ(f ) in the following way. The hypersurface Vftrop is a (n−1)-dimensional piecewise-linear complex which induces a polyhedral subdivision Ξ of Rn. We will call cells the elements of Ξ. Note that these cells have rational slopes. The n-dimensional cells of Ξ are the closures of the connected components of the complement of Vftrop in Rn. The lower dimensional cells of Ξ are contained in Vftrop and we will just say that they are cells of Vftrop.
Consider a cell ξ of Vftrop and pick a point x in the relative interior of ξ. Then the set Ix = {w ∈ Δ(f ) ∩ Zn | ∃ x ∈ Rn, ftrop(x) = x, w + val(cw)} is independent of x, and denote by Δξ the convex hull of this set. All together the polyhedra Δξ form a subdivision τ of Δ(f ) called the dual subdivision, and the cell Δξ is called the dual of ξ. Both subdivisions τ and Ξ are dual in the following sense. There is a one-to-one correspondence between Ξ and τ , which reverses the inclusion relations, and such that if Δξ ∈ τ corresponds to ξ ∈ Ξ then
1. dim ξ + dim Δξ = n,
2. the cell ξ and the polytope Δξ span orthogonal real affine spaces,
3. the cell ξ is unbounded if and only if Δξ lies on a proper face of Δ(f ).
2.2. A brief introduction to tropical geometry 30
Note that τ coincides with the regular subdivision of Definition 2.7 described in Subsection 2.2.1.
ˆ n × R be the convex hull of the points (w, ν(w)) with w ∈ W and ν(w) = Indeed, let Δ(f ) ⊂ R
ord(cw). Define
νˆ : Δ(f ) −→ R
x → ˆ min{y | (x, y) ∈ Δ(f )}.
Then, the the domains of linearity of νˆ form the dual subdivision τ .
Consider a facet (face of dimension n−1) ξ of Vftrop, then dim Δξ = 1 and we define the weight of ξ by w(ξ) := Card(Δξ ∩ Zn) − 1. Tropical varieties satisfy the so-called balancing condition. Since in Chapter 6, we only work with tropical curves in R2, we give here this property only for this case. We refer to [Mik06] for the general case. Let T ⊂ Rn be a tropical curve, and let v be a vertex of T . Let ξ1, . . . , ξl be the edges of T adjacent to v. Since T is a rational graph, each edge ξi has a primitive integer direction. If in addition we ask that the orientation of ξi defined by this vector points away from v, then this primitive integer vector is unique. Let us denote by uv,i this vector.
Proposition 2.11 (Balancing condition). For any vertex v, one has w(ξi)uv,i = 0. i=1
Intersection of tropical hypersurfaces
Consider polynomials f1, . . . , fk ∈ K[z1±1, . . . , zn±1]. For i = 1, . . . , k, let Δi ⊂ Rn (resp. Ti ⊂ Rn) denote the Newton polytope (resp. tropical curve) associated to fi. Recall that each tropical curve Ti defines a piecewise linear polyhedral subdivision Ξi of Rn which is dual to a convex polyhedral subdivision τi of Δi. The union of these tropical curves defines a piecewise-linear polyhedral subdivision Ξ of Rn. Any non-empty cell of Ξ can be written as ξ = ξ1 ∩ • • • ∩ ξk with ξi ∈ Ξi for i = 1, . . . , k. We require that ξ does not lie in the boundary of any ξi, thus any cell ξ ∈ Ξ can be uniquely written in this way. Denote by τ the mixed subdivision of the Minkowski sum Δ = Δ1 + • • • + Δk induced by the tropical polynomials f1, . . . , fk. Recall that any polytope σ ∈ τ comes with a privileged representation σ = σ1 + • • • + σk with σi ∈ τi for i = 1, . . . , k. The above duality-correspondence applied to the (tropical) product of the tropical polynomials gives rise to the following well-known fact (see [BB13] for instance).
Proposition 2.12. There is a one-to-one duality correspondence between Ξ and τ , which reverses the inclusion relations, and such that if σ ∈ τ corresponds to ξ ∈ Ξ, then
1. if ξ = ξ1 ∩ • • • ∩ ξk with ξi ∈ Ξi for i = 1, . . . , k, then σ has representation σ = σ1 + • • • + σk where each σi is the polytope dual to ξi.
2. dim ξ + dim σ = n,
3. the cell ξ and the polytope σ span orthonogonal real affine spaces,
4. the cell ξ is unbounded if and only if σ lies on a proper face of Δ.
Notation 2.13. In what follows, we denote such a σ by Δξ and we say that each polytope Δξ a mixed polytope of τ .
Definition 2.14. A cell ξ is transversal if it satisfies dim(Δξ) = dim(Δξ1 ) + • • • + dim(Δξk ), and it is non transversal if the previous equality does not hold.
Generalized Viro theorem and tropical reformulation
An important direction in real algebraic geometry is the construction of real algebraic hypersurfaces with prescribed topology (see [Ris92, Vir84] or [Vir89] for example). Central to these developments is a combinatorial construction due to O.Ya. Viro, which is based on regular triangulations of Newton polytopes. Using this technique, significant progress has been made in the study of low degree curves in the real projective plane (Hilbert’s 16th problem). Since Chapter 6 of this thesis concerns algebraic sets of dimension zero contained in (R>0)n, we only describe in this section how to use combinatorial patchworking in that orthant of Rn.
Following the description of B. Sturmfels [Stu94], we recall now Viro’s Theorem for hypersur-faces. Let W ⊂ Zn be a finite set of lattice points, and denote by Δ the convex hull of W. Assume that dim Δ = n and let ϕ : W → Z be any function inducing a regular triangulation τϕ of the integer convex polytope Δ (see Definition 2.7). Fix non-zero real numbers cw, w ∈ W. For each positive real number t, we consider a Laurent polynomial ft(z1, . . . , zn) = cwtϕ(w)zw. (2.2.1) w W
Let Bar(τϕ) denote the first barycentric subdivision of the regular triangulation τϕ. Each max-imal cell µ of Bar(τϕ) is incident to a unique point w ∈ W. We define the sign of a maximal cell µ to be the sign of the associated real number cw. The sign of any lower dimensional cell λ ∈ Bar(τϕ) is defined as follows:
+ if sign(µ) = + for all maximal cells µ containing λ,
sign(λ) := − if sign(µ) = − for all maximal cells µ containing λ,
0 otherwise.
Let Z+(τϕ, f ) denote the subcomplex of Bar(τϕ) consisting of all cells λ with sign(λ) = 0, and let V+(ft) denote the zero set of ft in the positive orthant of Rn. Denote by Int(Δ) the relative interior of Δ. Theorem 2.15 (Viro). For sufficiently small t > 0, there exists a homeomorphism (R>0)n → Int(Δ) sending the real algebraic set V+(ft) ⊂ (R>0)n to the simplicial complex Z+(τϕ, f ) ⊂ Int(Δ).
Naturally, a signed version of Theorem 2.15 holds in each of the 2n orthants (R>0)ǫ := {(x1, . . . , xn) ∈ (R∗)n | sign(xi) = ǫi for i = 1, . . . , n}, where ǫ ∈ {+, −}n. In fact, O. Viro proves a more general Theorem for Theorem 2.15, in which he defines a set that is homeomorphic to the the zero set V (ft) ⊂ Rn (not only the positive zero set V+(ft)) by means of gluing the zero sets of ft contained in all other orthants of Rn.
We now reformulate Theorem 2.15 using tropical geometry. We consider g := ft as a polynomial defined over the field of real generalized locally convergent Puiseux series, where each coefficient cwtϕ(w) ∈ RK∗ of g has only one term. Therefore coef(cwtϕ(w )) = cw, val(cwtϕ(w)) = −ϕ(w), and we associate to g a tropical hypersurface Vgtrop as defined in Subsection 2.2.2. Recall that Vgtrop induces a subdivision Ξg of Rn that is dual to τϕ. The tropical hypersurface Vgtrop is homeomorphic to the barycentric subdivision Bar(τϕ). Indeed, τϕ is a triangulation, and thus Bar(τϕ) becomes dual to τϕ in the sense of the duality described in Subsection 2.2.3.
We define for each n-cell ξ ∈ Ξg , dual to a 0-face (vertex) w of the triangulation τϕ, a sign ǫ(w) ∈ {+, −}, to be equal to the sign of cw.
Definition 2.16. The positive part, denoted by Vg,trop+, is the subcomplex of Vgtrop consisting of all (n − 1)-cells of Vgtrop that are adjacent to two n-cells of Vgtrop having different signs. A positive facet ξ+ is an (n − 1)-dimensional cell of Vg,trop+.
The following is a Corollary of Mikhalkin [Mik04] and Rullgard [Rul01] results, where they completely describe the topology of V (ft) using amoebas.
Theorem 2.17 (Mikhalkin, Rullgard). For sufficiently small t > 0, there exists a homeomorphism (R>0)n → Rn sending the zero set V+(ft) ⊂ (R>0)n to Vg,trop+ ⊂ Rn.
B. Sturmfels generalized Viro’s method for complete intersections in [Stu94]. We give now a tropical reformulation of one of the main Theorems of [Stu94].
Consider a system f1,t(z1, . . . , zn) = • • • = fk,t(z1, . . . , zn) = 0, (2.2.2) of k equations, where all ft,i are polynomial (2.2.1). For i = 1, . . . , k, we define as before gi := fi,t as a polynomial in RK[z1±1, . . . , zn±1]. Let V+(f1,t, . . . , fk,t) ⊂ (R>0)n denote the set of positive solutions of (2.2.2).
Transversal intersection points and discrete mixed volume
Assume now that the number of polynomials in (2.2.2) is equal to that of variables (i.e. k = n), and assume that the tropical hypersurfaces V trop, . . . , V trop intersect transversally. Then the trop trop g1 trop gn intersection set V+ (g1, . . . , gn) := Vgi,+ ∩ • • • ∩ Vgk ,+ is a (possibly empty) set of points in Rn. Each point p of V+trop(g1, . . . , gn) is expressed in a unique way as a transversal intersection ξ1,+ ∩ • • • ∩ ξn,+, where for i = 1, . . . , n, the cell ξi,+ ⊂ V trop is a positive cell. Theorem 2.18 gi,+ is a powerful tool for constructing polynomial systems (2.2.2) with many non-degenerate positive solutions.
A consequence of F. Bihan’s more general result [Bih14] is a bound on the number of positive mixed points for a system (2.2.2). For any number r of finite sets W1, . . . , Wr in Rn, and for any non-empty I ⊂ [r] = {1, 2, . . . , r}, write WI for the set of points i∈I wi over all wi ∈ Wi with i ∈ I. The associated discrete mixed volume of W1, . . . , Wr is defined as D(W1, . . . , Wr) =(−1)r−|I||WI |, (2.2.3) I⊂[r] where the sum is taken over all subsets I of [r] including the empty set with the convention that |W∅| = 1. Denote by Wi the support of gi for i = 1, . . . , n. Recall that the tropical hypersurfaces associated to g1, . . . , gn intersect transversally.
Theorem 2.19 (Bihan). volume D(W1, . . . , Wn). The number ♯{V trop ∩ • • • ∩ V trop} is less or equal to the discrete mixed g1 gn Obviously, we have ♯{V trop ∩ • • • ∩ V trop } ≤ ♯{V trop ∩ • • • ∩ V trop} g1,+ gn,+ g1 gn
Moreover, Theorem 1.4 of [Bih14] states that for any finite sets W1, . . . , Wr ⊂ Rn, we have D(W1, . . . , Wr) ≤ (|Wi| − 1). i∈[r]
Combining the latter result with Theorem 2.19 shows that Kushnirenko’s conjecture is true for polynomial systems constructed by the combinatorial patchworking method of Viro, or equivalently, for tropical polynomial systems given by transversal intersections of tropical hypersurfaces.
To our knowledge, we do not know if the discrete mixed volume bound is sharp for any poly-nomial system with n equations in n variables satisfying that the associated tropical hypersurfaces intersect transversally. An interesting direction to start, is to look at a system (2.2.2) such that all polynomials of (2.2.2) have the same support W. For example, when |W| = 4, then the bound of Theorem 2.19 is 3 and is sharp, see [Bih07].
When |W| = 5 and n = 2, we have D(W, W) = 6. We construct using combinatorial patch-working (Theorem 2.18) a polynomial system of two equations in two variables having a total of five distinct monomials and six non-degenerate solutions in (R>0)2. Thus proving that the bound of Theorem 2.19 is sharp when n = 2 and W1 = W2 = 5.
Reduced systems and non-transversal intersections
Theorem 2.18 is only adapted for the case where the tropical intersections are transverse. Therefore, we need other machinery to locate the valuations of positive solutions.
Types of non-transversal cells
In Chapter 6 of this thesis, we only work with tropical hypersurfaces in dimension two. Therefore, we classify the types of mixed cells ξ in the case where two tropical plane curves intersect non- transversally at a cell ξ. Let ξ denote the relative interior of ξ. Note that ξ = ξ if ξ is a point.
Assume that ξ is non-transversal, we distinguish three types for such ξ.
• A cell ξ is of type (I) if dim ξ = dim ξ1 = dim ξ2 = 1.
• A cell ξ is of type (II) if one of the cells ξ1, or ξ2 is a vertex, and the other cell is an edge.
• A cell ξ is of type (III) if ξ1 and ξ2 are vertices of the corresponding tropical curves.
Figure 2.2: The three types of non-transversal intersection cells.
Reduced systems
Recall that for an element a(t) ∈ K∗, we denote by coef(a(t)) the non-zero coefficient corresponding to the term of α(t) with the smallest exponent of t.
Definition 2.20. Let f = w∈Δ(f )∩Z 2 cwzw be a polynomial in K[z±1 , z±1 ] let ξ denote a cell of Vftrop. The reduced polynomial f|ξ ∈ C[z1±1, z2±1] of f a polynomial defined as with cw ∈ K∗, and with respect to ξ is f|ξ = coef(cw)zw, w∈Δξ ∩W where W is the support of f .
We extend this definition to the following. Consider a system f1(z) = f2(z) = 0, (2.2.4) with f1, f2 in K[z1±1, z2±1] defined as above. Assume that the intersection set T1 ∩ T2 of the tropical curves T1 and T2 is non-empty, and consider a mixed cell ξ ∈ T1 ∩ T2. As explained in Subsection 2.2.4, the mixed cell ξ is written as ξ1 ∩ ξ2 for some unique ξ1 ∈ T1 and ξ2 ∈ T2.
Definition 2.21. The reduced system of (2.2.4) with respect to ξ is the system f1|ξ1 = f2|ξ2 = 0, with fi|ξi is the reduced polynomial of fi with respect to ξi for i = 1, 2.
Let W1 and W2 denote the supports of f1 and f2 respectively, and write f1(z) = av zv and f2(z) = bwzw. v∈W1 w∈W2
The following result also generalizes to a polynomial system defined on the same field with n equations in n variables.
Proposition 2.22. If the system (2.2.4) has a solution (α, β) ∈ (K∗)2 such that Val(α, β) ∈ ξ, then (coef(α), coef(β)) is a real solution of the reduced system f1|Δξ 1 = f2|Δξ 2 = 0. (2.2.5)
Proof. Assume that (2.2.4) has a solution (α, β) ∈ (K∗)2 such that Val(α, β) ∈ ξ. Since Val(α, β) belongs to the relative interior of each of ξ1 and ξ2, we have max{ Val(α, β), v + val(av ), v ∈ W1 \ (W1 ∩ Δξ1 )} < Val(α, β), v + val(av ) for v ∈ W1 ∩ Δξ1 and max{ Val(α, β), w +val(bw), w ∈ W2 \(W2 ∩Δξ2 )} < Val(α, β), w +val(bw) for w ∈ W2 ∩Δξ2 .
Consequently, since ord = − val, we have M := − Val(α, β), v− val(av ) and N := − Val(α, β), w− val(bw) are the orders of f1(α, β) and f2(α, β) respectively. Therefore, replacing (z1, z2) by tord(α)z1, tord(β)z2 in (2.2.4), such a system becomes f1 tord(α)z1, tord(β)z2 = tM v W1 ∩Δξ 1 coef(av )zv + g1(z) , (2.2.6) f2 tord(α)z1, tord(β)z2 = tN w W2 ∩Δξ 2 coef(bw)zw + g2(z) , where all the coefficients of the polynomials g1 and g2 of RK[z1±1, z2±1] have positive orders. Since (α, β) is a non-zero solution of (2.2.5), the system (2.2.6) has a non-zero solution (α0, β0) with ord(α0) = ord(β0) = 0 and Coef(α, β) = Coef(α0, β0). It follows that taking t > 0 small enough, we get that Coef(α0, β0) is a non-zero solution of coef(av )zv = coef(bw)zw = 0. v W1∩Δξ1 w W2∩Δξ2
Note that Proposition 2.22 holds true for any type of tropical intersection cell ξ. However, the other direction does not always hold true when ξ is of type (I). Recall that a solution (α, β) ∈ (K∗)2 is positive if (α, β) ∈ (RK∗>0)2.
Proposition 2.23. Assume that dim ξ = 0 and that all solutions of (2.2.4) are non-degenerate. If the reduced system of (2.2.4) with respect to ξ has a non-degenerate solution (ρ1, ρ2) ∈ (R∗>0)2, then (2.2.4) has a non-degenerate solution (α, β) ∈ (RK∗>0)2 such that Val(α, β) = ξ and Coef(α, β) = (ρ1, ρ2).
Proof. E. Brugall´e and L. L´opez De Medrano showed in [BLdM12, Proposition 3.11] (see also [Kat09, Rab12, OP13] for more details for higher dimension and more exposition relating toric varieties and tropical intersection theory) that the number of solutions of (2.2.4) with valuation ξ is equal to the mixed volume MV(Δξ1 , Δξ2 ) of ξ1 and ξ2 (recall that Δξ = Δξ1 + Δξ2 ). Since we assumed that (2.2.4) has only non-degenerate solutions in (K∗)2, we get MV(Δξ1 , Δξ2 ) distinct solutions of the system (2.2.4) in (K∗) 2 with given valuation ξ. By Proposition 2.22, if f1(z) = f2(z) = 0 and Val(z) = ξ, then Coef(z) is a solution of the reduced system of (2.2.4) with respect to ξ. The number of solutions of the reduced system in (C∗)2 is MV(Δξ1 , Δξ2 ). Assuming that this reduced system has MV(Δξ1 , Δξ2 ) distinct solutions in (C∗)2, we obtain that the map z Coef(z) induces a bijection from the set of solutions of (2.2.4) in (K∗) 2 with valuation ξ onto the set of solutions in (C∗)2 of the reduced system of (2.2.4) with respect to ξ.
If z is a solution of (2.2.4) in (K∗)2 with Val(z) = ξ and Coef(z) ∈ (R∗)2, then z ∈ (RK∗)2 since otherwise, z, z¯ would be two distinct solutions of (2.2.4) in (K∗ \ RK∗)2 such that Val(z) = Val(¯z) = ξ and Coef(z) = Coef(¯z).
Table of contents :
1 Introduction
1.1 Univariate polynomials
1.2 Sparse polynomial systems
1.2.1 Polyhedral bounds
1.2.2 Fewnomial bounds
1.3 Results prior to this thesis
1.3.1 Around Khovanskii’s bound
1.3.2 Using combinatorial patchworking
1.3.3 Systems supported on a circuit
1.3.4 Around Kuschnirenko’s conjecture
1.3.5 Around a polynomial-fewnomial conjecture
1.4 Results of the thesis
1.4.1 Chapter 3: Intersecting a sparse plane curve and a line
1.4.2 Chapter 4: Positive intersection points of a trinomial and a t-nomial curves
1.4.3 Chapter 5: Characterization of circuits supporting polynomial systems with the maximal number of positive solutions
1.4.4 Chapter 6: Constructing polynomial systems with many positive solutions
2 Preliminaries
2.1 A brief introduction to real dessins d’enfant
2.2 A brief introduction to tropical geometry
2.2.1 Polytopes and subdivisions
2.2.2 Tropical polynomials and hypersurfaces
2.2.3 Tropical hypersurfaces and subdivisions
2.2.4 Intersection of tropical hypersurfaces
2.2.5 Generalized Viro theorem and tropical reformulation
2.2.6 Reduced systems and non-transversal intersections
3 Intersecting a sparse plane curve and a line
3.1 Preliminary results
3.2 Proof of Theorem 3.1
3.3 Optimality
4 Positive intersection points of a trinomial and a t-nomial curves
4.1 Introduction and statement of the main results
4.2 Proof of Theorem 4.1
4.3 Proof of Theorem 4.2
4.3.1 Reduction to a simpler case
4.3.2 Analysis of dessins d’enfant
4.3.3 End of the proof of Theorem 4.2
4.4 The case of two trinomials: proof of Theorem 4.3
4.4.1 Proof of Proposition 4.33
4.4.2 End of proof of Theorem 4.3
5 Characterization of circuits
5.1 Technical preamble
5.2 Proof of the “only if” direction of Theorem 5.1
5.3 Proof of the “if” direction of Theorem 5.1
6 Constructing polynomial systems
6.1 Statement of the main results
6.1.1 For normalized systems
6.1.2 Transversal intersection points
6.2 Non-transversal intersection components of type (I)
6.3 Base fans and tropical intersections
6.4 Preliminary case-by-case analysis for n = k = 2
6.4.1 Approximation polynomials for type-(I) intersections
6.4.2 Reduced systems for type-(II) intersections
6.4.3 Reduced systems for type-(III) intersections at the origin
6.4.4 Type-(III) intersections outside the origin
6.5 Proof of Theorem 6.1
6.5.1 First part of Theorem 6.48
6.5.2 Construction: second part of Theorem 6.48
6.6 Proof of Theorem 6.3 (part 1)
6.6.1 First case: 0 < α < β
6.6.2 The case α = 0 < β
6.6.3 The case α < 0 < β
6.6.4 The case α < 0 and β = 0
6.6.5 The case α < β < 0
6.6.6 The case α = β < 0
6.7 Proof of Theorem 6.3 (part 2)
6.7.1 The case 0 < α ≤ β
6.7.2 The case α < 0 < β
6.7.3 The case α ≤ β < 0
7 Introduction (en Fran¸cais)
Bibliography
