Related papers: On quadratic curves over finite fields
We compute the number of points over finite fields of some algebraic varieties related to cluster algebras of finite type. More precisely, these varieties are the fibers of the projection map from the cluster variety to the affine space of…
We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…
We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are…
With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by…
We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…
In this paper we study division algebras over the function fields of curves over $\Q_p$. The first and main tool is to view these fields as function fields over nonsingular $S$ which are projective of relative dimension 1 over the $p$ adic…
We transfer the algebro-geometric method of construction of solutions of the discrete KP equation to the finite field case. We emphasize role of the Jacobian of the underlying algebraic curve in construction of the solutions. We illustrate…
In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
We study plane curves over finite fields whose tangent lines at smooth $\mathbb{F}_q$-points together cover all the points of $\mathbb{P}^2(\mathbb{F}_q)$.
We are interested in shapes of real algebraic curves in the plane and regions surrounded by them: they are named refined algebraic domains by the author. As characteristic finite sets, we consider points contained in two curves and the sets…
In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of…
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses…
We consider quadric surface fibrations over curves, defined over algebraically closed and finite fields. Our goal is to understand, in geometric terms, spaces of sections for such fibrations. We analyze varieties of maximal isotropic…
Consider a finite scheme of length l contained in a smooth quadric surface over the complex numbers. We determine the number of linearly independent curves passing through the scheme, of degree at least l - 1.
There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What…
Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
By studying $\mathbb{A}^1$-curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete…
We present some results about the number of rational points on a certain family of curves defined over a finite field. In a small number of cases the curves have more rational points than expected. Fibonacci numbers make an appearance, as…
We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…