Variation of the canonical height for polynomials in several variables

TL;DR

This paper investigates how the canonical height varies in families of polynomial endomorphisms over number fields, showing that under certain conditions, the height difference is bounded, refining previous asymptotic results.

Contribution

It improves existing results by proving the canonical height differs from a Weil height by a bounded amount in specific polynomial families with invariant hyperplanes.

Findings

Canonical height differs from Weil height by little-o in general.

In special cases, the difference is bounded.

Results refine understanding of height variation in polynomial families.

Abstract

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 Weil height on the base by little-o of a Weil height. In the case where f is a family with an invariant hyperplane, whose restriction to this invariant hyperplane is isotrivial, we improve this by showing that the canonical height along a section differs from a Weil height on the base by a bounded amount.