Arithmetic derivatives through geometry of numbers

2021/06/30 by Pasten, Hector
#14A23 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11J97 #Secondary: 11H06

paper · doi:10.48550/arxiv.2106.16165

Abstract

We define certain arithmetic derivatives on 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.

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