相关论文: Determination of hyperovals by lines through a few…

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a…

Let Fq be a finite field with q=8 or q at least 16. Let S be a smooth cubic surface defined over Fq containing at least one rational line. We use a pigeonhole principle to prove that all the rational points on S are generated via tangent…

Newton's quadrilateral theorem can be phrased as follows. If H is a circle that is tangent to the four extended sides of a non-parallelogram quadrilateral Q, the center of H lies on the Newton line of Q. We prove that the theorem remains…

A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chv\'atal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, this…

Let $f$ be an $E$-function (in Siegel's sense) not of the form $e^{\beta z}$, $\beta \in \overline{\mathbb{Q}}$, and let $\log$ denote any fixed determination of the complex logarithm. We first prove that there exists a finite set $S(f)$…

A pseudo-hyperoval of a projective space $\PG(3n-1,q)$, $q$ even, is a set of $q^n+2$ subspaces of dimension $n-1$ such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is…

The Sylvester-Gallai theorem says that for any finite set of non-collinear points in $\R^2$, there is some line passing through exactly two points of the set. Over the complex numbers, this theorem fails: there are finite configurations…

For a variety of finite groups $\mathbf H$, let $\overline{\mathbf H}$ denote the variety of finite semigroups all of whose subgroups lie in $\mathbf H$. We give a characterization of the subsets of a finite semigroup that are pointlike…

We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.

In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $…

In this paper, we study the problem of finding the largest possible set of s points and s lines in a projective plane of order q, such that that none of the s points lie on any of the s lines. We prove that s <= 1+(q+1)(\sqrt{q}-1). We also…

We show that, if a $n$-vertex triangulation $T$ of maximum degree $\Delta$ has a dual that contains a cycle of length $\ell$, then $T$ has a non-crossing straight-line drawing in which some \emph{collinear set} of $\Omega(\ell/\Delta^4)$…

We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively…

We prove that a bivariate polynomial f with exactly t non-zero terms, restricted to a real line {y=ax+b}, either has at most 6t-4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether…

We prove the following: (a) Let X be a smooth, codimension two subvariety of P6. If X lies on a hyperquintic or if deg(X)<74, then X is a complete intersection. (b) Let X be a smooth, subcanonical threefold in P5. If X lies on a…

Let $f_1,\dots,f_m$ be polynomials in $n$ variables with coefficients in a finite field $\mathbb{F}_q$. We estimate the number of points $\underline{x}$ in $\mathbb{F}_q^n$ such that each value $f_i(\underline{x})$ is a nonzero square in…

For each integer $s\geq 1$, we present a family of curves that are $\mathbb{F}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s = 2$, we give necessary and sufficient conditions for…

We prove discrete Helly-type theorems for pseudohalfplanes, which extend recent results of Jensen, Joshi and Ray about halfplanes. Among others we show that given a family of pseudohalfplanes $\cal H$ and a set of points $P$, if every…

We present a deterministic 2^O(t)q^{(t-2)(t-1)+o(1)} algorithm to decide whether a univariate polynomial f, with exactly t monomial terms and degree <q, has a root in F_q. A corollary of our method --- the first with complexity sub-linear…

Let $p$ denote the characteristic of ${\mathbb F}_q$, the finite field with $q$ elements. We prove that if $q$ is odd then an arc of size $q+2-t$ in the projective plane over ${\mathbb F}_q$, which is not contained in a conic, is contained…