Related papers: Inhomogeneous quadratic congruences
It has been established on many occasions that the set of quotients of prime numbers is dense in the set of positive real numbers. More recently, it has been proved that the set of quotients of primes in the Gaussian integers is dense in…
We discuss the rigidity problem for Mori fibrations on del Pezzo surfaces of degree 1, 2 and 3 over ${\mathbb P}^1$ and formulate the following conjecture: such a del Pezzo fibration $V/{\mathbb P}^1$ is birationally rigid if and only if…
We solve categorical Torelli problem for quartic del Pezzo surfaces. That is, we prove that a del Pezzo surface of degree $4$ can be canonically reconstructed from its Kuznetsov component, which is the orthogonal subcategory to the…
We obtain asymptotic bounds on the number of natural numbers less than $X$ satisfying $\gcd{\left(n,\lfloor P(n) \rfloor\right)}=1$, under some diophantine conditions on the coefficient of $x$ in $P$, and show that the density of such…
We compute invariants of quadratic forms associated to orthogonal hypergeometric groups of degree five. This allows us to determine some commensurabilities between these groups, as well as to say when some thin groups cannot be conjugate to…
We classify del Pezzo surfaces with 1/3(1,1) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake), we tabulate their biregular invariants, we give good model…
Given two distinct newforms with real Fourier coefficients, we show that the set of primes where the Hecke eigenvalues of one of them dominate the Hecke eigenvalues of the other has density at least 1/16. Furthermore, if the two newforms do…
We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height…
We study the density of complex zeros of a system of real random SO($m+1$) polynomials in several variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of…
The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…
In this paper we contribute towards the classification of partially symmetric tensors in $\mathbb{F}_q^3\otimes S^2\mathbb{F}_q^3$, $q$ even, by classifying planes which intersect the Veronese surface $\mathcal{V}(\mathbb{F}_q)$ in at least…
In this article, we give the classification of normal del Pezzo surfaces of rank one with at most log canonical singularities containing the affine plane defined over an algebraically non-closed field of characteristic zero. As an…
The intersection of a quadric and a cubic surface in 3-space is a canonical curve of genus 4. It has 120 complex tritangent planes. We present algorithms for computing real tritangents, and we study the associated discriminants. We focus on…
A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we…
We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate…
We classify del Pezzo surfaces with Picard number is equal to one and with four log terminal singular points.
We study the surface quasi-geostrophic equation with an irregular spatial perturbation $$ \partial_{t }\theta+ u\cdot\nabla\theta = -\nu(-\Delta)^{\gamma/2}\theta+ \zeta,\qquad u=\nabla^{\perp}(-\Delta)^{-1}\theta, $$ on…
We call an integer a \emph{near-square} if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers $a \geq 3$ by $u_{0}(a)=0$, $u_{1}(a)=1$ and…
In this paper, we study the K-stability of del Pezzo surfaces with a single quotient singularity whose minimal resolution admits exactly two exceptional curves \(E_1\) and \(E_2\) with \(E_{1}^2=-n\), \(E_{2}^2=-m\) for \(n,m\geq 2\).
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < \epsilon$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the…