Related papers: The density of rational points on non-singular hyp…

S. Lang conjectured in 1974 that a hyperbolic algebraic variety defined over a number field has only finitely many rational points, and its analogue over function fields. We discuss the Nevanlinna-Cartan theory over function fields of…

For any prime $p\ge 5$, we show that generic hypersurface $X_p\subset\mathbb{P}^p$ defined over $\mathbb{Q}$ admits a non-trivial rational dominant self-map of degree $>1$, defined over $\bar{\mathbb{Q}}$. A simple arithmetic application of…

We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a…

For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither zero nor one. We quantify this statement, following work by V. Kolyada, and obtain the…

We study density thresholds that force a measurable set $E\subseteq\mathbb{R}^d$ to contain all sufficiently large similar copies of every $n$-point configuration. We prove a lower bound of the form $1-O((\log n)/n)$, which matches the…

For a nonempty compact set D of R we determine the maximal possible dimension of a subspace X of polynomial functions of degree at most m which possesses a positive bases (where positivity is understood on D). The exact value of this…

In this article we prove a conjecture formulated by A.B. Soerensen in 1991 on the maximal number of $\mathbb{F}_{q^2}$-rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree $d \le q.$

Let $X$ be a smooth projective algebraic variety over a number field $k$ and $P$ in $X(k)$. In 2007, the second author conjectured that, in a precise sense, if rational points on $X$ are dense enough, then the best rational approximations…

In this note we consider Diophantine equations of the form \begin{equation*} a(x^p-y^q) = b(z^r-w^s), \quad \mbox{where}\quad \frac{1}{p}+\frac{1}{q}+\frac{1}{r}+\frac{1}{s}=1, \end{equation*} with even positive integers $p,q,r,s$. We show…

In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case…

For natural numbers $n$ and $l > d \geq 2$, let $ES_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in $\mathbb{R}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in…

It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+\epsilon)$-spanner with maximum degree $\tilde{O}(\epsilon^{-d+1})$ and with lightness $\tilde{O}(\epsilon^{-d})$; 2)…

For a nonzero rational number $q$, a rational $D(q)$-$n$-tuple is a set of $n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$ is a square for all $1 \leqslant i < j \leqslant n$. We investigate for which $q$…

The set of primes where a hypergeomeric series with rational parameters is $p$-adically bounded is known by [10] to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density…

We design and analyze an algorithm for computing rational points of hypersurfaces defined over a finite field based on searches on "vertical strips", namely searches on parallel lines in a given direction. Our results show that, on average,…

Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$ are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra.…

We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic…

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results…

Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid…