Related papers: Strong approximation and descent

We study strong approximation of the equation N_{L/k}(x) = \prod_{i=1}^n p_i(t) where L/k is a finite extension of number fields and p_i(t)'s are distinct irreducible polynomials over k. We prove this equation satisfies strong approximation…

We give a new version of the open descent theory of Harari and Skorobogatov. As an application of the new version, we prove that some algebraic varieties satisfy strong approximation.

For rational points on algebraic varieties defined over a number field $K$, we study the behavior of the property of weak approximation with Brauer-Manin obstruction under extension of the ground field. We construct K-varieties accompanied…

Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three…

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…

Let K/Q be a field extension of finite degree and let P(t) be a polynomial over Q that splits into linear factors over Q. We show that any smooth model of the affine variety defined by the equation N_{K/Q} (k) = P(t) satisfies the Hasse…

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 give local conditions at the infinite places of a number field K ensuring that the intersection of n quadrics in projective N-space over K, N >> n, satisfies weak approximation.

We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical…

In this paper we study two types of descent in the category of Berkovich analytic spaces: flat descent and descent with respect to an extension of the ground field. Quite surprisingly, the deepest results in this direction seem to be of the…

We introduce the notion of {\it approximation type} for the partial, and in certain cases the total description of extensions of a given valuation from a field $K$ to the rational function field $K(x)$. To every extension, a unique…

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…

Let K be a non-archimedean field, and let f in K(z) be a rational function of degree d>1. If f has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that f is conjugate…

We study the use of approximate Lagrange multipliers and discrete actions in solving convex optimisation problems. We observe that descent, which can be ensured using a wide range of approaches (gradient, subgradient, Newton, etc.), is…

We introduce a variant of the $k$-nearest neighbor classifier in which $k$ is chosen adaptively for each query, rather than supplied as a parameter. The choice of $k$ depends on properties of each neighborhood, and therefore may…

Consider strong approximation for algebraic varieties defined over a number field $k$. Let $S$ be a finite set of places of $k$ containing all archimedean places. Let $E$ be an elliptic curve of positive Mordell-Weil rank and let $A$ be an…

We introduce and study coordinate-wise powers of subvarieties of $\mathbb{P}^n$, i.e. varieties arising from raising all points in a given subvariety of $\mathbb{P}^n$ to the $r$-th power, coordinate by coordinate. This corresponds to…

Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…

Let us consider a generalized Artin-Schreier algebraic function field extension $F$ of the rational function field $\F_{p^n}(x)$ defined over the finite field extension $K=\F_{p^n}$ of the prime field $\F_p$. We assume that $K$ is…

Descent polynomials and peak polynomials, which enumerate permutations with given descent and peak sets respectively, have recently received considerable attention. We give several formulas for $q$-analogs of these polynomials which refine…