9 citations · 12 across the 10 of their papers we have counts for
14 papers
A Geometric Criterion for Degeneracy in the Elekes-Szabó Theorem
Mehdi Makhul
The Elekes-Szabó theorem establishes that an irreducible algebraic hypersurface contains few grid points unless it exhibits a specific group-related structure. Identifying t…
Irreducible components of sets of points in the plane that satisfy distance conditions
Niels Lubbes, Mehdi Makhul, Josef Schicho +1
For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between th…
On Galois groups of type-1 minimally rigid graphs
Mehdi Makhul, Josef Schicho, Audie Warren
For every graph that is mimimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edg…
On sets of points in general position that lie on a cubic curve in the plane and determine lines that can be pierced by few points
Mehdi Makhul, Rom Pinchasi
Let be a set of points in general position in the plane. Let be a set of points disjoint from such that for every the line through and contains…
Normality of the Thue-Morse function for finite fields along polynomial values
Mehdi Makhul, Arne Winterhof
Let be the finite field of elements, where is a power of the prime , and be an ordered basis of …
The Elekes-Szabó Problem and the Uniformity Conjecture
Mehdi Makhul, Oliver Roche-Newton, Sophie Stevens +1
In this paper we give a conditional improvement to the Elekes-Szabó problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}…
No perfect triangle is isosceles
Mehdi Makhul
A perfect triangle is a triangle with rational sides, medians, and area. In this article, we use a similar strategy due to Pocklington to show that if is a perfect triangle, th…
The Spherical Kakeya Problem in Finite Fields
Mehdi Makhul, Audie Warren, Arne Winterhof
We study subsets of the -dimensional vector space over the finite field , for odd , which contain either a sphere for each radius or a sphere for each first coo…
On the number of perfect triangles with a fixed angle
Mehdi Makhul
Richard Guy asked the following question: can we find a triangle with rational sides, medians, and area? Such a triangle is called a \emph{perfect triangle} and no example has been…
Constructions for the Elekes-Szabó and Elekes-Rónyai problems
Mehdi Makhul, Oliver Roche-Newton, Audie Warren +1
We give a construction of a non-degenerate polynomial and a set of cardinality such that $\left|Z(F)\cap (A \times A \times A) \right| \gg n^{\frac{…
An application of Bertini Theorem
Mehdi Makhul, Josef Schicho
Given an irreducible variety over a finite field, the density of hypersurfaces of varying degree intersecting in an irreducible subvariety is , by a result of Charle…
A family of four-variable expanders with quadratic growth
Mehdi Makhul
We prove that if is a polynomial of constant degree that does not divide , then for any finite set \[ |X| \gg_d…
Probabilities of incidence between lines and a plane curve over finite fields
Mehdi Makhul, Josef Schicho, Matteo Gallet
We study the probability for a random line to intersect a given plane curve, defined over a finite field, in a given number of points defined over the same field. In particular, we…
On sets defining few ordinary circles
Aaron Lin, Mehdi Makhul, Hossein Nassajian Mojarrad +3
An ordinary circle of a set of points in the plane is defined as a circle that contains exactly three points of . We show that if is not contained in a line or a cir…