Related papers: Estimating heights using auxiliary functions

This paper employs the add-and-subtract technique of the auxiliary receiver approach to establish a new upper bound for the distributed hypothesis testing problem. This new bound has fewer assumptions than the upper bound proposed by Rahman…

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…

In this study we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Then we find height bounds in the polynomial ring over algebraic numbers to test…

The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an…

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. We utilize -- and reprove -- D. Anick's fundamental result on…

Let K be a number field, X/K a curve, and f/X a family of endomorphisms of projective N-space. It follows from a result of Call and Silverman that the canonical height associated to the family f, evaluated along a section, differs from a…

A complex number alpha is said to satisfy the height reducing property if there is a finite subset F of the ring Z of the rational integers such that Z[alpha]=F[alpha]. This problem of finding F has been considered by several authors,…

This paper is devoted to present new error bounds of regularized gap functions for polynomial variational inequalities with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved…

In the first part we study deviation of a polynomial from its mathematical expectation. This deviation can be estimated from above by Carbery--Wright inequality, so we investigate estimates of the deviation from below. We obtain such…

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive…

We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to…

We give upper bounds for volume of sublevel sets of real polynomials. Our method is to combine a version of global Lojasiewicz inequality with some well known estimate on volume of tubes around real algebraic sets. Some applications to…

We give upper bounds for the number of integral solutions of bounded height to a system of equations $f_i(x_1,\ldots,x_n) = 0$, $1 \leq i \leq r$, where the $f_i$ are polynomials with integer coefficients. The estimates are obtained by…

In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions, such as being totally real or p-adic, improving on…

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit…

We give a new proof of a result of DiPippo and Wan for counting points of bounded height on projective spaces over global function fields. The new proof adapts the geometry of numbers arguments used by Schanuel in the number field case.

The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that for any sufficiently generic algebraic subvariety of a semiabelian $\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height of the points…

We give upper and lower bounds for weighted Chebyshev and residual polynomials on subsets of the real line. As an application, we prove a Szeg\H{o}-type theorem in the setting of Parreau--Widom sets.

We derive explicit expressions for $q$-orthogonal polynomials arising in the enumeration of area-weighted Dyck paths with restricted height.

Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius…