Arithmetic derivatives through geometry of numbers

TL;DR

This paper constructs specific arithmetic derivatives on integers using Geometry of Numbers, showing their existence is linked to the $abc$ Conjecture, thus connecting number theory, derivatives, and geometric methods.

Contribution

It introduces a new class of arithmetic derivatives respecting Leibniz rule and additive properties, and establishes their existence and size bounds unconditionally, relating them to the $abc$ Conjecture.

Findings

Existence of arithmetic derivatives with controlled size proven using Geometry of Numbers.

Small derivatives imply a version of the $abc$ Conjecture.

Existence of small derivatives is equivalent to the $abc$ Conjecture.

Abstract

We define certain arithmetic derivatives on $\mathbb{Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a suitable non-degeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the $abc$ Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the $abc$ Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighties, relating Geometry of Numbers in arithmetic to derivatives in function fields and Nevanlinna theory. In addition, our construction formalizes the widespread intuition that the $abc$ Conjecture should be related to arithmetic derivatives of some sort.