Abstract. We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t − 7. This improves a result by M. Avendano. Furthermore, we prove that this bound is sharp for t = 3 with the help of Grothendieck's dessins d'enfant.
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 non-transversal intersections of tropical curves.
We study the problem of counting real simple rational functions ϕ with prescribed ramification data (i.e. a particular class of oriented real Hurwitz numbers of genus 0). We introduce a signed count of such functions which is independent of the position of the branch points, thus providing a lower bound for the actual count (which does depend on the position). We prove (non-)vanishing theorems for these signed counts and study their asymptotic growth when adding further simple branch points. The approach is based on Itenberg and Zvonkine (Comment Math Helv 93(2), 441-474, 2018) which treats the polynomial case.
A polynomial system with n equations in n variables supported on a set W ⊂ R n of n + 2 points has at most n + 1 non-degenerate positive solutions. Moreover, if this bound is reached, then W is minimally affinely dependent, in other words, it is a circuit in R n . For any positive integer number n, we determine all circuits W ⊂ R n which can support a polynomial system with n+1 non-degenerate positive solutions. Restrictions on such circuits W are obtained using Grothendieck's real dessins d'enfant, while polynomial systems with n + 1 non-degenerate positive solutions are constructed using Viro's combinatorial patchworking.
A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is called its coupler curve. To each calligraph we uniquely assign a vector consisting of three integers. This vector bounds the degrees and geometric genera of irreducible components of the coupler curve. A graph, that up to rotations and translations admits finitely many, but at least two, realizations into the plane for almost all edge length assignments, is a union of two calligraphs. We show that this number of realizations is equal to a certain inner product of the vectors associated to these two calligraphs. As an application we obtain an improved algorithm for counting numbers of realizations, and by counting realizations we characterize invariants of coupler curves.
For any two integers k, n, 2 ≤ k ≤ n, let f : (C * ) n → C k be a generic polynomial map with given Newton polytopes. It is known that points, whose fiber under f has codimension one, form a finite set C1(f ) in C k . For maps f above, we show that C1(f ) is empty if k ≥ 3, we classify all Newton polytopes contributing to C1(f ) = ∅ for k = 2, and we compute |C1(f )|.
The discriminant of a polynomial map between two spaces is an object central to many problems in mathematics. Standard methods for computing it rely on elimination techniques which can be inefficient. This paper concerns polynomial maps on the two-dimensional torus defined over the field of Puiseux series with complex coefficients. We present a combinatorial procedure for computing the tropical curve of the discriminant of such maps that satisfy some mild genericity conditions. Thanks to the advances made in tropical geometry, our results give rise to a new tool for investigating the discriminant of a complex polynomial map on the plane, and for computing its Newton polytope.
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