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Boulos El Hilany

Publications and source records attributed to Boulos El Hilany.

17 recordsLinked to original sources
arXiv2025

Around the topological classification problem of polynomial maps: A survey

The study of the topology of polynomial maps originates from classical questions in affine geometry, such as the Jacobian Conjecture, as well as from works of Whitney, Thom, and Mather in the 1950-70s on diffeomorphism types of smooth maps. During that period, Thom came up with a famous construction of a one-dimensional family of real polynomial maps all sharing the same degree, but in which every polynomial map has its unique topological type. According to his convention, the topological type of a map is preserved precisely when it is composed with homeomorphisms on both source and target spaces. Thom also conjectured that for each pair $(n,d)$, any family of $n$--variate, degree--$d$ (complex, or real) polynomial functions has at most finitely-many topological types. Soon after, a collection of results by several mathematicians throughout the 1970s and 1980s settled this conjecture, and solved its subsequent generalization to polynomial maps. In this survey, we outline the historical context and highlight a range of significant works from the 1950s to the present day that lead to the current state of the art in the study of polynomial maps' topology. The focal point of this survey is to shed some light on the ensuing classification problem of topological types of polynomial maps. The presentation is achieved by making a gentle introduction to several other prominent questions in affine geometry, all of which are recounted through the lens of this classification problem.

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arXiv2024

Counting positive intersection points of a trinomial and a $\mathbf{T}$-nomial curves via Groethendieck's dessin d'enfant

We consider real polynomial systems $f=g=0$ in two variables where $f$ has $t\geq 3$ monomial terms and $g$ has $3$ monomials terms. We prove that the number of positive isolated solutions of such a system does not exceed $3\cdot 2^{t-2} - 1$. This improves the bound $2^t - 2$ obtained in [T.-Y. Li, J.-M. Rojas and X. Wang, 2003]. This also refines for $t=4,\ldots,9$ the bound $2t^3/3 + 5t$ obtained in [P. Koiran, N. Portier and S. Tavenas, 2015]. Our proof is based on a delicate analysis of the Groethendieck's dessin d'enfant associated to some rational function determined by the system. For $t=3$, it was shown in [T.-Y. Li, J.-M. Rojas and X. Wang, 2003] that the sharp bound is five, and if this bound is reached, then the Minkowski sum of the associated Newton triangles is an hexagon. A further analysis of Groethendieck's dessin d'enfant allows us to show that if the bound five is reached, then there exist two consecutive edges of the hexagon which are translate of two consecutive edges of one Newton triangle.

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arXiv2024

Improved fewnomial upper bounds from Wronskians and dessins d'enfant

We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^α(1-x)^β P - Q$, has at most $°P +°Q + 2$ roots in the interval $]0,~1[$. As a consequence, we obtain an upper bound on the number of positive solutions to a real polynomial system $f=g=0$ in two variables where $f$ has three monomials terms, and $g$ has $t$ terms. The approach we adopt for tackling this Fewnomial bound relies on the theory of Wronskians, which was used in Koiran et.\ al.\ (J.\ Symb.\ Comput., 2015) for producing the first upper bound which is polynomial in $t$.

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arXiv2024

Stratification of Projection Maps From Toric Varieties

We prove a combinatorial version of Thom's Isotopy Lemma for projection maps applied to any complex or real toric variety. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all examples we produced, our resulting algorithm outperforms known general purpose methods in Helmer and Nanda (FoCM, 2022), and Dinh and Jelonek (DCG, 2021) for computing map-stratifications.

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arXiv2024

Bounds on the infimum of polynomials over a generic semi-algebraic set using asymptotic critical values

We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our bounds depend on the degree, the bitsize of $f$, and the polynomials defining $\mathscr{X}$, and are single exponential with respect to the number of variables. They generalize the single exponential bounds from Jeronimo, Perrucci, and Tsigaridas (SIAM Journal on Optimization, 23(1):241--255, 2013) for the minimum of a polynomial function on a compact connected component of a basic closed semi-algebraic set. The tools that we use allow us to obtain specialized bounds and dedicated algorithms for two large families of polynomial optimization problems in which the optimum value might not be attained. The first family forms a dense set of real polynomial functions with a fixed collection of Newton polytopes; we provide the best approximation yet for the bifurcation set, which contains the optimal value, and we deduce an effective method for computations. As for the second family, we consider any unconstrained polynomial optimization problem; we present more precise bounds, together with a better bit complexity estimate of an algorithm to compute the optimal value.

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arXiv2024

The tropical non-properness set of a polynomial map

We study some discrete invariants of Newton non-degenerate polynomial maps $f : \mathbb{K}^n \to \mathbb{K}^n$ defined over an algebraically closed field of Puiseux series $\mathbb{K}$, equipped with a non-trivial valuation. It is known that the set $\mathcal{S}(f)$ of points at which $f$ is not finite forms an algebraic hypersurface in $\mathbb{K}^n$. The coordinate-wise valuation of $\mathcal{S}(f)\cap (\mathbb{K}^*)^n$ is a piecewise-linear object in $\mathbb{R}^n$, which we call the tropical non-properness set of $f$. We show that the tropical polynomial map corresponding to $f$ has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of $f$. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of $\mathcal{S}(f)$ in terms of multivariate resultants.

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arXiv2024

The polyhedral type of a polynomial map on the plane

Two continuous maps $f, g : \mathbb{C}^2\to\mathbb{C}^2$ are said to be topologically equivalent if there exist homeomorphisms $φ,ψ:\mathbb{C}^2\to\mathbb{C}^2$ satisfying $ψ\circ f\circφ= g$. It is known that there are finitely many topologically non-equivalent polynomial maps $\mathbb{C}^2\to\mathbb{C}^2$ with any given degree $d$. The number $T(d)$ of these topological types is known only whenever $d=2$. In this paper, we describe the topology of generic complex polynomial maps on the plane using the corresponding pair of Newton polytopes and establish a method for constructing topologically non-equivalent maps of degree $d$. We furthermore provide a software implementation of the resulting algorithm, and present lower bounds on $T(d)$ whenever $d=3$ and $d=4$.

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arXiv2023

Coupler curves of moving graphs and counting realizations of rigid graphs

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.

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arXiv2023

Computing the non-properness set of real polynomial maps in the plane

We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call \emph{the Jelonek set}, is a subset of $\mathbb{K}^2$ where a dominant polynomial map $f:\mathbb{K}^2\to\mathbb{K}^2$ is not proper; $\mathbb{K}$ could be either $\mathbb{C}$ or $\mathbb{R}$. Unlike all the previously known approaches we make no assumptions on $f$ whenever $\mathbb{K} = \mathbb{R}$; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in Maple.

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arXiv2022

The tropical discriminant of a polynomial map on a plane

The discriminant of a polynomial map is central to problems from affine geometry and singularity theory. Standard methods for characterizing it rely on elimination techniques that can often be ineffective. This paper concerns polynomial maps on the two-dimensional torus defined over a field of Puiseux series. We present a combinatorial procedure for computing the tropical curve of the discriminant of maps determined by generic polynomials with given supports. Our results enable one to compute the Newton polytope of the discriminant of complex polynomial maps on the plane.

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arXiv2021

Counting isolated points outside the image of a polynomial map

We consider a generic family of polynomial maps $f:=(f_1,f_2):\mathbb{C}^2\rightarrow\mathbb{C}^2$ with given supports of polynomials, and degree $ d(f):=\max (deg f_1, deg f_2)$. We show that the (non-) properness of maps $f$ in this family depends uniquely on the pair of supports and that the set of isolated points in $\mathbb{C}^2\setminus f(\mathbb{C}^2)$ has a size of at most $6 d(f)$. This improves an existing upper bound $(d(f) - 1)^2$ proven by Jelonek. Moreover, for each $n\in\mathbb{N}$, we construct a dominant map $f$ above, with $d(f) = 2n+2$, and having $2n$ isolated points in $\mathbb{C}^2\setminus f(\mathbb{C}^2)$. Our proofs are constructive and can be adapted to a method for computing isolated missing points of $f$. As a byproduct, we describe those points in terms of singularities of the bifurcation set of $f$.

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arXiv2020

A note on polynomial maps having fibers of maximal dimension

For any two integers $k,n$, $2\leq k\leq n$, let $f:(\mathbb{C}^*)^n\rightarrow\mathbb{C}^k$ 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 $C_1(f)$ in $\mathbb{C}^k$. For maps $f$ above, we show that $C_1(f)$ is empty if $k\geq 3$, we classify all Newton polytopes contributing to $C_1(f)\neq \emptyset$ for $k=2$, and we compute $|C_1(f)|$.

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arXiv2019

Signed counts of real simple rational functions

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 that is invariant under change of the branch locus, thus providing a lower bound for the actual count (which does depend on such change). 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 the works of Itenberg and Zvonkine (arXiv:1609.05219) which treat the polynomial case.

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arXiv2019

Describing the Jelonek set of polynomial maps via Newton polytopes

Let $\K=\C$, or $\R$, and $S_f$ be the set of points in $\K^n$ at which a polynomial map $f:\K^n\rightarrow\K^n$ is non-proper. Jelonek proved that $S_f$ is a semi-algebraic set that is ruled by polynomial curves, with $\dim S_f\leq n-1$, and provided a method to compute $S_f$ for $\K = \C$. However, such methods do not exist for $\K = \R$. In this paper, we establish a straightforward description of $S_f$ for a large family of non-proper maps $f$ using the Newton polytopes of the polynomials appearing in $f$. Thus resulting in a new method for computing $S_f$ that works for $\K=\R$, and highlights an interplay between the geometry of polytopes and that of $S_f$. As an application, we recover some of Jelonek's results, and provide conditions on (non-)properness of $f$. Moreover, we discover another large family of maps $f$ whose $S_f$ has dimension $n-1$ (even for $\K=\R$), satisfies an explicit stratification, and weak smoothness properties. This novel description allows our tools to be extended to all non-proper maps.

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arXiv2017

Constructing polynomial systems with many positive solutions using tropical geometry

The number of positive solutions of a system of two polynomials in two variables defined in 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. Tropical geometry is a powerful tool to construct polynomial systems with many positive solutions. The classical combinatorial patchworking method arises when the tropical hypersurfaces intersect transversally. In this paper, we prove that a system as above constructed using this method has at most 6 positive solutions. We also show that this bound is sharp. Moreover, using non-transversal intersections of tropical curves, we construct a system as above having 7 positive solutions.

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arXiv2016

Characterization of circuits supporting polynomial systems with the maximal number of positive solutions

A polynomial system with $n$ equations in $n$ variables supported on a set $\mathcal{W}\subset\mathbb{R}^n$ of $n+2$ points has at most $n+1$ non-degenerate positive solutions. Moreover, if this bound is reached, then $\mathcal{W}$ is minimally affinely dependent, in other words, it is a circuit in $\mathbb{R}^n$. For any positive integer number $n$, we determine all circuits $\mathcal{W}\subset\mathbb{R}^n$ which can support a polynomial system with $n+1$ non-degenerate positive solutions. Restrictions on such circuits $\mathcal{W}$ 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.

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