Related papers: Rational points near curves and small nonzero |x^3…

In this paper, we continue the study of three close factorizations of an integer and correct a mistake of a previous result. This turns out to be related to lattice points close to the center point $(\sqrt{N}, \sqrt{N})$ of the hyperbola $x…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic…

We study the exact counting problem for all lattice rectangles contained in the square $[0,n)\times[0,n)$, including non-axis-parallel ones. Starting from the standard parametrization by a primitive direction $(u,v)$ and two side lengths,…

In relation to Hall's conjecture, a new algorithm is presented to search for small k = |x^3 -y^2| values. Thirteen new values of k < x^(1/2) are reported.

We present a computable reformulation of Ruelle's linear response formula for chaotic systems. The new formula, called Space-Split Sensitivity or S3, achieves an error convergence of the order ${\cal O}(1/\sqrt{N})$ using $N$ phase points.…

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of…

In this note we present a fast algorithm that finds for any $r$ the number $N_r$ of $\mathbb{F}_{q^r}$ rational points on a smooth absolutely irreducible curve $C$ defined over $\mathbb{F}_{q}$ assuming that we know $N_1,\cdots,N_g$, where…

In this work, we consider the rational points on elliptic curves over finite fields F_{p}. We give results concerning the number of points on the elliptic curve y^2{\equiv}x^3+a^3(mod p)where p is a prime congruent to 1 modulo 6. Also some…

We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a…

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-cuts) that defines the convex hull of the integer…

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm (so-called LLL) has been…

The nearest lattice point problem in $\mathbb{R}^n$ is formulated in a distributed network with $n$ nodes. The objective is to minimize the probability that an incorrect lattice point is found, subject to a constraint on inter-node…

We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming…

We approach the Max-3-Cut problem through the lens of maximizing complex-valued quadratic forms and demonstrate that low-rank structure in the objective matrix can be exploited, leading to alternative algorithms to classical semidefinite…

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound…

Estimating pose from given 3D correspondences, including point-to-point, point-to-line and point-to-plane correspondences, is a fundamental task in computer vision with many applications. We present a complete solution for this task,…

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] for $C^3$ non-degenerate planar curves. With this goal in mind, here for the first…

A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a…

In the numerical linear algebra community, it was suggested that to obtain nearly optimal bounds for various problems such as rank computation, finding a maximal linearly independent subset of columns (a basis), regression, or low-rank…