Constructing polynomial systems with many positive solutions using tropical geometry
Abstract
The number of positive solutions to a system of two polynomials in two variables defined over the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. The main result of this paper is the construction of a system as above having 7 positive solutions. This is achieved using tools developed in tropical geometry. When the corresponding tropical hypersurfaces intersect transversally, one can easily estimate the positive solutions to the system using the classical combinatorial patchworking for complete intersections. We apply this generalization to construct a system as above having 6 positive solutions. We also show that this bound is sharp. Consequently, our main result is proved using nontransversal intersections of tropical curves.
Keywords
Real algebraic geometry Tropical geometry Solving polynomial systems Fewnomial theoryMathematics Subject Classification
14P99 14T05 13P15 14P251 Introduction and statement of the main results
This paper concerns one of the first cases where the sharp upper bound on the number of nondegenerate positive solutions to (1.1) is still unknown. Namely, with respect to the above notations, our main result involves a system (1.1) satisfying \(n=k=2\).
Theorem 1.1
There exists a real system (1.1) of two polynomials in two variables supported on a set \(\mathcal {W}\) of five distinct points in \({\mathbb {Z}}^2\) and having seven nondegenerate positive solutions.
It was proven in [6] that a sharp bound to such a system (of type \(n=k=2\) for short) is not greater than 15. On the other hand, the best examples had only 5 nondegenerate positive solutions. The first such published system, made by Haas [9], is a construction of two real bivariate trinomials, after which other similar examples of systems having 5 positive solutions followed in [7]. The authors in the latter paper also showed that such systems (consisting of two real bivariate trinomials) are rare in the following sense. They proved that if we pick a point at random in the discriminant variety of coefficient spaces of polynomial systems composed of two bivariate trinomials with fixed exponent vectors, this point has probability of order \(10^{9}\) to be located in a chamber (connected component of the complement) containing systems with the maximal number five of positive solutions.
Proposition 1.2
There exist two plane tropical curves \(T_1\) and \(T_2\) defined by equations containing a total of five monomials and which have six transversal intersection points. Moreover, each such point belongs to a positive facet of \(T_1\) and a positive facet of \(T_2\).
Due to Theorem 3.2, the construction made for proving the latter result also gives a construction of a real polynomial system of type \(n=k=2\) that has six positive solutions. Furthermore it is clear from Theorem 3.3 and Lemma 3.4 that one cannot hope to improve the result in Proposition 1.2 when restricting to polynomial systems of type \(n=k=2\) with tropical curves intersecting transversally. Consequently, in order to obtain a better construction, we consider real parametrized polynomial systems (1.4) of type \(n=k=2\), having only nondegenerate solutions, and whose tropical curves \(T_1\) and \(T_2\) intersect in a nonempty set that does not consist entirely of transversal points.
A consequence of an important result due to Kapranov [10] is that the set \(T_1\cap T_2\) contains the image under the valuation map (see Sect. 2.1) of the solutions to (1.4). The strategy for proving Theorem 1.1 is to deduce, from the combinatorial types of the connected components of \(T_1\cap T_2\), the number of nondegenerate positive solutions to (1.4) that map to \(T_1\cap T_2\). Namely, we decompose those connected components to linear pieces \(\xi \) which can either have dimension zero or one. Then, to each such \(\xi \), we associate a certain polynomial subsystem of (1.4), called reduced real systems with respect to \(\xi \) (see [8, Chapter 2.2.6]). All together, those reduced systems approximate all the nondegenerate parametrized solutions to (1.4) (see [11, 17, 18] and [5]). We adapt this approach to our setting by considering a particular type of parametrized nondegenerate solutions \((\alpha _1(t),\alpha _2(t))\), which we also call positive (i.e. the firstorder terms of \(\alpha _1(t)\) and \(\alpha _2(t)\) have positive coefficients). This refinement constitutes an important benefit concerning our construction. Specifically, when \(t>0\) is small enough, the system (1.4) produces a real system of type \(n=k=2\) having a set of nondegenerate positive solutions that is in a bijective correspondence with the set of the nondegenerate positive solutions to (1.4).
In [8, Chapter 6.4], a casebycase analysis was made, for a family of systems (1.4) of type \(n=k=2\), to identify the few classes of candidates of such systems that have more than six positive solutions. The construction described in the present paper is based on one such candidate.
This paper is organized as follows. We introduce in Sect. 2 some basic notions of tropical geometry. In Sect. 3, we give a description of the tropical reformulation of Viro’s Patchworking Theorem and its generalization followed by the proof of Proposition 1.2. Finally, Sect. 4 is devoted to the proof of Theorem 1.1.
2 A brief introduction to tropical geometry
We state in this section some of the wellknown facts about tropical geometry (see [4], and the references therein), much of the exposition and notations in this section are taken from [1, 5, 19].
Definition 2.1

\(\cup _{i\in I}\Delta _i=\Delta \), and

if \(i,j\in I\), then if the intersection \(\Delta _i\cap \Delta _j\) is nonempty, it is a common face of the polytope \(\Delta _i\) and the polytope \(\Delta _j\).
Definition 2.2
Let \(\Delta \) be a convex polytope in \({\mathbb {R}}^n\) and let \(\tau \) denote a polyhedral subdivision of \(\Delta \) consisting of convex polytopes. We say that \(\tau \) is regular if there exists a continuous, convex, piecewiselinear function \(\varphi :~\Delta \rightarrow {\mathbb {R}}\) such that the polytopes of \(\tau \) are exactly the domains of linearity of \(\varphi \).
2.1 Tropical polynomials and hypersurfaces
Notation 2.1
Let \({{\mathrm{coef}}}(a(t))\) denote the coefficient \(\alpha _{{{\mathrm{val}}}(a(t))}\) of the first term of a(t) following the increasing order of the exponents of t. We extend \({{\mathrm{coef}}}\) to a map \({{\mathrm{Coef}}}:~{\mathbb {K}}^n\rightarrow \mathbb {R}^n\) by taking \({{\mathrm{coef}}}\) coordinatewise, i.e. \({{\mathrm{Coef}}}(a_1(t),\ldots , a_n(t))=({{\mathrm{coef}}}(a_1(t)),\ldots , {{\mathrm{coef}}}(a_n(t)))\).
We use the convention that for any \(s\in {\mathbb {K}}\), we have \({{\mathrm{coef}}}(s)=0\Leftrightarrow s=0\) and \({{\mathrm{val}}}(s)=\infty \Leftrightarrow s=0\).
An element \(a(t)=\sum _{r\in R} \alpha _rt^r\) of \({\mathbb {K}}\) is said to be real if \(\alpha _r\in \mathbb {R}\) for all r, and positive if a(t) is real and \({{\mathrm{coef}}}(a(t))>0\). Denote by \(\mathbb {RK}\) (resp. \(\mathbb {RK}_{>0}\)) the subfield of \({\mathbb {K}}\) composed of real (resp. positive) series. Since elements of \({\mathbb {K}}\) are convergent for \(t>0\) small enough, an algebraic variety over \({\mathbb {K}}\) (resp. \(\mathbb {RK}\)) can be seen as a oneparametric family of algebraic varieties over \({\mathbb {C}}\) (resp. \(\mathbb {R}\)).
Theorem 2.2
(Kapranov) A tropical hypersurface \(V_f^{{{\mathrm{trop}}}}\) of a polynomial f defined over an algebraically closed field is the corner locus of the Legendre transform \(\mathcal {L}(\nu _f)\).
For a positive integer l, we define the tropical summation of \(x_1,\ldots ,x_l\in {\mathbb {R}}\cup \{\infty \}\) as their usual maximum \(\max (x_1,\ldots ,x_l)\), and the tropical multiplication as their usual sum \(\sum _{i=1}^lx_i\). A tropical polynomial is a polynomial in \({\mathbb {R}}[x_1,\ldots ,x_n]\), where the addition and multiplication 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.
Example 2.3
2.2 Tropical hypersurfaces and subdivisions
Keeping with the same notation as above, the tropical hypersurface \(V_f^{{{\mathrm{trop}}}}\) is an \((n1)\)dimensional piecewiselinear complex which produces a polyhedral subdivision \(\Xi \) of \({\mathbb {R}}^n\). This subdivision induces a regular subdivision \(\tau \) of the Newton polytope \(\Delta (f)\) of f in the following way (see [1, Section 3] for more details). The elements of \(\Xi \), called cells, have rational slopes. The ndimensional cells of \(\Xi \) are the closures of the connected components of the complement of \(V_f^{{{\mathrm{trop}}}}\) in \({\mathbb {R}}^n\). The lower dimensional cells of \(\Xi \) are contained in \(V_f^{{{\mathrm{trop}}}}\) and we will just say that they are cells of \(V_f^{{{\mathrm{trop}}}}\).
Consider now polynomials \(f_1,\ldots ,f_r~\in {\mathbb {K}}[z_1^{\pm 1},\ldots ,z_n^{\pm 1}]\). For \(i=1,\ldots ,r\), let \(\mathcal {W}_i\subset {\mathbb {Z}}^n\) (resp. \(\Delta _i\subset {\mathbb {R}}^n\), \(T_i\subset {\mathbb {R}}^n\)) denote the support (resp. Newton polytope, tropical hypersurface) associated to \(f_i\).
Proposition 2.4
 (1)
if \(\xi =\xi _1\cap \cdots \cap ~\xi _r\) with \(\xi _i\in \Xi _i\) for \(i=1,\ldots ,r\), then \(\sigma \) has representation \(\sigma = \sigma _1 +\cdots +\sigma _r\) where each \(\sigma _i\) is the polytope dual to \(\xi _i\).
 (2)
\(\dim \xi + \dim \sigma = n\),
 (3)
the cell \(\xi \) and the polytope \(\sigma \) span orthonogonal real affine spaces,
 (4)
the cell \(\xi \) is unbounded if and only if \(\sigma \) lies on a proper face of \(\Delta \).
Definition 2.3
A cell \(\xi \) is transversal if it satisfies \(\dim (\Delta _\xi )=\dim (\Delta _{\xi _1})+\cdots +\dim (\Delta _{\xi _r})\), and it is non transversal if the previous equality does not hold.
3 First construction: transversal case
3.1 Generalized Viro theorem and tropical reformulation
Viro’s combinatorial patchworking [23, 24] can be interpreted via the socalled “dequantization” method (see [25]). In what follows, we use tropical hypersurfaces to formulate this description.
Definition 3.1
The positive part, denoted by \(V_{g,+}^{{{\mathrm{trop}}}}\), is the subcomplex of \(V_g^{{{\mathrm{trop}}}}\) consisting of all \((n1)\)cells of \(V_g^{{{\mathrm{trop}}}}\) that are adjacent to two ncells of \(\Xi _g\) having different signs (see the left part of Fig. 2 for an example). A positive facet \(\xi _+\) is an \((n1)\)dimensional cell of \(V_{g,+}^{{{\mathrm{trop}}}}\).
Theorem 3.1
(Viro) Let \(f_t\in \mathbb {RK}[z_1^{\pm 1},\ldots ,z_n^{\pm 1}]\) be a polynomial (3.1), denote by \(V_+(f_t)\subset ({\mathbb {R}}_{>0})^n\) the intersection of the zero set of \(f_t\) with the positive orthant of \({\mathbb {R}}^n\), and by \(V_{g,+}^{{{\mathrm{trop}}}}\subset \mathbb {R}^n\) the positive part of the tropical hypersurface corresponding to \(g:=f_t\). Then for sufficiently small \(t>0\), there exists a homeomorphism \(({\mathbb {R}}_{>0})^n\rightarrow {\mathbb {R}}^n\) sending \(V_+(f_t)\subset ({\mathbb {R}}_{>0})^n\) to \(V_{g,+}^{{{\mathrm{trop}}}}\subset {\mathbb {R}}^n\).
In fact, O. Viro proves a more general version of Theorem 3.1 by similarly describing the whole zero set \(V(f_t)\subset {\mathbb {R}}^n\) (not only \(V(f_t)\cap ({\mathbb {R}}_{>0})^n\)). Later, B. Sturmfels generalized Viro’s method for complete intersections in [22]. We give now a tropical reformulation of one of the main Theorems of [22].
Theorem 3.2
(Sturmfels) Let \(f_{1,t},\ldots ,f_{r,t}\in \mathbb {RK}[z_1^{\pm 1},\ldots ,z_n^{\pm 1}]\) be r polynomials of the form (3.1), and denote by \(V_+(f_{1,t},\ldots ,f_{r,t})\) their common real zerolocus in the positive orthant of \({\mathbb {R}}^n\). For \(i=1,\ldots ,r\), let \(V_{g_i,+}^{{{\mathrm{trop}}}}\) denote the positive part of the tropical hypersurface \(V_{g_i}^{{{\mathrm{trop}}}}\subset {\mathbb {R}}^n\) corresponding to \(g_i:=f_{i,t}\), and assume that \(V_{g_1}^{{{\mathrm{trop}}}}\cap \cdots \cap V_{g_r}^{{{\mathrm{trop}}}}\) is a union of transversal cells (see Definition 2.3). Then for sufficiently small \(t>0\), there exists a homeomorphism \(({\mathbb {R}}_{>0})^n\rightarrow {\mathbb {R}}^n\) sending the real algebraic set \(V_+(f_{1,t},\ldots ,f_{r,t})\subset ({\mathbb {R}}_{>0})^n\) to the intersection \(V_{g_1,+}^{{{\mathrm{trop}}}}\cap \cdots \cap V_{g_r,+}^{{{\mathrm{trop}}}}\subset {\mathbb {R}}^n\).
Similarly to Viro’s work, B. Sturmfels generalizes Theorem 3.2 for the zero set
\(V(f_{1,t},\ldots ,f_{r,t})\subset {\mathbb {R}}^n\) (see [22, Theorem 5]).
3.2 Tropical transversal intersection points for bivariate polynomials
Theorem 3.3
(Bihan) If the intersection locus \(V_{g_1}^{{{\mathrm{trop}}}}\cap V_{g_2}^{{{\mathrm{trop}}}}\) of the tropical curves corresponding to the polynomials in (3.2) consists only of transversal points, then \(V_{g_1}^{{{\mathrm{trop}}}}\cap V_{g_2}^{{{\mathrm{trop}}}}\) is less or equal to the discrete mixed volume \(D(\mathcal {W}_1,\mathcal {W}_2)\) of the corresponding supports.
When \(\mathcal {W}_1\cup \mathcal {W}_2=4\), then the bound of Theorem 3.3 is 3, and is sharp (see [2]). However, we do not know if the discrete mixed volume bound is sharp for any polynomial system with 2 equations in 2 variables satisfying that the associated tropical curves intersect transversally.
3.3 Restriction to the case \(n=k=2\)
We need the following result.
Lemma 3.4
If \(\mathcal {W}_1\) and \(\mathcal {W}_2\) are two subsets of \({\mathbb {Z}}^2\) satisfying \(\mathcal {W}_1\cup \mathcal {W}_2=5\), then the discrete mixed volume \(D(\mathcal {W}_1,\mathcal {W}_2)\) does not exceed six.
Proof
Assume now that \(\mathcal {W}_1\cap \mathcal {W}_2=3\). If \(\mathcal {W}_1=3\) and \(\mathcal {W}_2=5\) (the case where \(\mathcal {W}_1=5\) and \(\mathcal {W}_2=3\) is treated similarly), then (3.5) gives \(\mathcal {W}_1+\mathcal {W}_2\le 12\), and thus we get \(D(\mathcal {W}_1,\mathcal {W}_2)\le 5\). Finally, if \(\mathcal {W}_1=\mathcal {W}_2=4\), then \(\mathcal {W}_1+\mathcal {W}_2\le 13\), from which we deduce that \(D(\mathcal {W}_1,\mathcal {W}_2)\le 6\), and the statement is proved. \(\square \)
We finish this section by proving Proposition 1.2.
Proof of Proposition 1.2
4 Second construction: nontransversal case

A cell \(\xi \) is of type (I) if \(\dim \xi =\dim \xi _1=\dim \xi _2=1\).

A cell \(\xi \) is of type (II) if one of the cells \(\xi _1\), or \(\xi _2\) is a vertex, and the other cell is an edge.

A cell \(\xi \) is of type (III) if \(\xi _1\) and \(\xi _2\) are vertices of the corresponding tropical curves.
4.1 Construction

all \(a_i,b_j\) are real generalized Puiseux series in t and having valuation zero,

the support of (4.1) is a subset of \({\mathbb {Z}}^2\) with no three points belonging to a line, and

the integers \(m_1,n_2\) are positive and the numbers \(\alpha ,\beta \) are real.
First, Theorem 1.16 of [8] implies that if \(\alpha \ne \beta \) or \(\alpha =\beta <0\), then (4.1) has at most six positive solutions. This was done in [8, Chapter 6.6 – 6.7] via a casebycase analysis of the coefficients and exponent vectors of the system (4.1). Namely, for each realizable intersection locus \(T_1\cap T_2\), one can provide an upper bound \(N(\xi )\) on the number of positive solutions to (4.1) with valuation in any linear piece \(\xi \) of \(T_1\cap T_2\). Then, proving that not all \(N(\xi )\) can attain their maximal possible value at once, gives the upper bound six if either \(\alpha \ne \beta \) or \(\alpha =\beta <0\).
On the other hand, the special case where \(\alpha =\beta =0\) was not treated in [8], for the latter approach does not provide an upper bound better than fifteen, which is previouslyknown (see [6]) and remains unclear whether it is sharp or not.
We assume henceforth that \(\alpha =\beta >0\). From the equations appearing in (4.1), it is easy to deduce that, since \(\alpha ,\beta > 0\), the intersection locus of tropical curves \(T_1\) and \(T_2\) contains a nontransversal intersection point of type (III) at the origin \(v_0=(0,0)\). The 3valent vertex \(v_0\) is adjacent to three 1dimentional linear pieces of \(T_1\cap T_2\), one of which has a vertical direction we denote by \(\mathsf {E}_0\) (Fig. 5 shows the intersection locus in red).

The common vertex \(v_0\) is the valuation of five positive solutions,

the transversal intersection point p is the valuation of one positive solution, and

the common vertical edge \(\mathsf {E}_0\) contains the valuation q of one positive solution.
4.1.1 Solutions with valuation \(v_0\)
Lemma 4.1
If (4.2) has a solution \((\lambda _1,\lambda _2)\in ({\mathbb {K}}^*)^2\) such that \({{\mathrm{Val}}}(\lambda _1,\lambda _2)=v_0\), then by taking \(t>0\) small enough, the point \({{\mathrm{Coef}}}(\lambda _1,\lambda _2)\in ({\mathbb {C}}^*)^2\) becomes a solution to the system (4.4).
Proof
The following result gives an explicit description of the solutions in \((\mathbb {RK}_{>0})^2\) to (4.2), and thus of those to (4.1), with valuation at \(v_0\).
Proposition 4.2
Assume that all solutions to (4.2) are nondegenerate. If the system (4.4) has a solution \((\rho _1,\rho _2)\in ({\mathbb {R}}_{>0})^2\), then (4.2) has a nondegenerate solution \((\lambda _1,\lambda _2)\in (\mathbb {RK}_{>0})^2\) such that \({{\mathrm{Val}}}(\lambda _1,\lambda _2)=v_0\) and \({{\mathrm{Coef}}}(\lambda _1,\lambda _2)=(\rho _1,\rho _2)\).
Proof
Recall that \(v_0\) is a tropical intersection point of type (III) for the polynomial system (4.2). Denote by \(\xi _1\) (resp. \(\xi _3\)) the 3valent vertex of the tropical curve associated to the first (resp. second) equation of (4.2) such that \(v_0=\xi _1\cap \xi _3\), and denote by \(\Delta _{\xi _1}\) (resp. \(\Delta _{\xi _3}\)) its dual triangle in \({\mathbb {R}}^2\). Note that \(\Delta _{\xi _1}\) (resp. \(\Delta _{\xi _3}\)) is the Newton triangle of the first (resp. second) equation of (4.4). E. Brugallé and L. López De Medrano showed in [5, Proposition 3.11] (see also [11, 17, 18] for more details for higher dimension and more exposition relating toric varieties and tropical intersection theory) that the number of solutions of (4.2) with valuation at \(v_0\) is equal to the mixed volume \({{\mathrm{MV}}}(\Delta _{\xi _1},\Delta _{\xi _3})\) of \(\Delta _{\xi _1}\) and \(\Delta _{\xi _3}\) (recall that \(\Delta _{v_0}=\Delta _{\xi _1}+\Delta _{\xi _3}\)). Since we assumed that (4.2) has only nondegenerate solutions in \(({\mathbb {K}}^*)^2\), we get \({{\mathrm{MV}}}(\Delta _{\xi _1},\Delta _{\xi _3})\) distinct solutions of the system (4.2) in \(({\mathbb {K}}^*)^2\) with valuation \(v_0\). By Lemma 4.1, if \((\lambda _1,\lambda _2)\) is a solution to (4.2) and \({{\mathrm{Val}}}(\lambda _1,\lambda _2)=v_0\), then \({{\mathrm{Coef}}}(\lambda _1,\lambda _2)\) is a solution to (4.4). The number of solutions in \(({\mathbb {C}}^*)^2\) to (4.4) is \({{\mathrm{MV}}}(\Delta _{\xi _1},\Delta _{\xi _3})\). Assuming that the latter system has \({{\mathrm{MV}}}(\Delta _{\xi _1},\Delta _{\xi _3})\) distinct solutions in \(({\mathbb {C}}^*)^2\), we obtain that the map \(z:=(z_1,z_2)~\mapsto {{\mathrm{Coef}}}(z_1,z_2)\) induces a bijection from the set of solutions to (4.2) in \(({\mathbb {K}}^*)^2\) with valuation at \(v_0\) onto the set of solutions in \(({\mathbb {C}}^*)^2\) to (4.4).
If z is a solution to (4.2) in \(({\mathbb {K}}^*)^2\) with \({{\mathrm{Val}}}(z)=v_0\) and \({{\mathrm{Coef}}}(z)\in ({\mathbb {R}}^*)^2\), then \(z\in (\mathbb {RK}^*)^2\) since otherwise, \(z,\bar{z}\) would be two distinct solutions to (4.2) in \(({\mathbb {K}}^*{\setminus }\mathbb {RK}^*)^2\) such that \({{\mathrm{Val}}}(z)={{\mathrm{Val}}}(\bar{z})=v_0\) and \({{\mathrm{Coef}}}(z)={{\mathrm{Coef}}}(\bar{z})\). \(\square \)
4.1.2 Choosing the coefficients
Proposition 4.3
Proof
First, we define a univariate function f such that for some constant c, the equation \(f=c\) has the same number of solutions in ]0, 1[ as that of positive solutions to (4.6). We write the first equation of (4.6) as \(z_2 = x^{k}(1  x)^{l}\), where \(x:=z_1^{m_1}\), \(\displaystyle k=m_2/(m_1 n_2)\) and \(l =1/n_2\). It is clear that \(z_1,z_2>0\Leftrightarrow x\in I_0:=]0,1[\).
Assuming that this is the case, it is only left to compute \({{\mathrm{coef}}}(c_2)\). Plotting the function \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\), \(x\mapsto f(x)\), we get that the graph of f has four critical points contained in \(I_0\) with critical values situated above the xaxis. Moreover, this graph intersects transversally the line \(\{y=0.36008\}\) in five points with the first coordinates belonging to \(I_0\). Taking \({{\mathrm{coef}}}(c_2)=\,0.36008\), the equation \(f(x)=0.36008\) has five nondegenerate positive solutions in \(I_0\), and we have finished the proof. \(\square \)
4.1.3 Choosing the exponent vectors
In what follows, we show how to find \((m_i,n_i)\in {\mathbb {Z}}^2\) for \(i=1,2,3,4\), satisfying the equalities in (4.7) so that (4.6) has five nondegenerate positive solutions, and (4.2) has seven.
Recall that \(T_1\) and \(T_2\) denote the tropical curves associated respectively to the first and second equations of the original system (4.1). From the discussion in [8, Chapter 6.5.1.1], imposing \(\alpha =\beta =\gamma _2<\gamma _0\) makes it impossible for the vertical edge \(\mathsf {E}_0\) of the intersection locus \(T_1\cap T_2\) to contain the valuation of more than one positive solution to (4.1) (see [8, Chapters 6.2 and 6.4.1] for the exposition on the method used to approximate the positive solutions to (4.1), where the tropical curves intersect at components of type (I)).
Assume in what follows that \(\mathsf {E}_0\) contains the valuation of one positive solution to (4.1). This implies that both \(n_3\) and \(n_4\) are positive (see [8, Chapter 6.5.2]), and thus all the vertices of both Newton polytopes of the polynomials appearing in (4.1) have nonnegative second coordinate (recall that \(n_2\) is assumed to be positive). Since in addition we have \(\alpha >0\), the edge \(\mathsf {E}_0\) in the tropical picture of \(T_1\cap T_2\), is a halfray with direction \((0,\,1)\) (see Fig. 5 for an example).
From the discussion in [8, Chapter 6.5.1], if (4.1) has more than six positive solutions, then \(v_0\) is the only nontransversal intersection point of type (III). Namely, an extra intersection point v of type (III) imposes additional restrictions on the support of (4.1) (and consequently, on the support of (4.4)). These restrictions force v to be the valuation of at most one positive solution, they additionally prohibit both (4.4) having five positive solutions, and \(T_1\cap T_2\) having transversal intersections.
We also deduce from equalities in (4.7) that \(l_4>l_3\) and \(k_4>k_3\), and thus \(n_4>n_3\) and \((m_4m_3)n_2  (n_4n_3)m_2>0\). Fixing \((m_3,n_3)\) in the region \(B_1\), we obtain that \((m_4,n_4)\) belongs to the triangle \(B_{1,1}\) depicted in Fig. 6.
All these restrictions impose that there exists a transversal intersection point of \(T_1\) and \(T_2\). Moreover, since \({{\mathrm{coef}}}(b_4)<0\) [see (4.7)], \({{\mathrm{coef}}}(a_3)=1\) [from (4.6)] and \({{\mathrm{coef}}}(a_0)={{\mathrm{coef}}}(b_0)=1\), the intersection point p is the valuation of a positive solution to (4.1). This follows from [8, Proposition 6.27].
The constant \({{\mathrm{coef}}}(c_0)\) should be a positive number so that (4.1) would have a positive solution with valuation in \(\mathsf {E}_0\). This constant can take any positive value, and for computational reasons we choose it to be 0.36008.
4.1.4 A software computation
Notes
Acknowledgements
Open access funding provided by Max Planck Society. I am very grateful to Frédéric Bihan for fruitful discussions and guidance. I would like to thank PierreJean Spaenlehauer for computations that approximated the positive solutions to the system that was constructed to prove Theorem 1.1. I also thank the anonymous referee for helpful remarks and suggestions on earlier versions of this paper.
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