Upper bounds on the heights of polynomials and rational fractions from their values

TL;DR

This paper establishes upper bounds on the height of univariate polynomials and rational fractions over number fields based on their values at small integers, improving existing bounds with more evaluation points.

Contribution

It provides new tighter bounds on the height of polynomials and rational fractions by analyzing their values at multiple points, extending classical interpolation bounds.

Findings

Tighter bounds on polynomial heights from multiple evaluation points

Improved estimates for rational fractions over number fields

Analysis of height bounds based on evaluation at small integers

Abstract

Let $F$ be a univariate polynomial or rational fraction of degree $d$ defined over a number field. We give bounds from above on the absolute logarithmic Weil height of $F$ in terms of the heights of its values at small integers: we review well-known bounds obtained from interpolation algorithms given values at $d+1$ (resp. $2d+1$) points, and obtain tighter results when considering a larger number of evaluation points.