Arithmetic derivatives through geometry of numbers

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dc.contributor.authorPasten, Hector
dc.date.accessioned2025-01-20T22:03:06Z
dc.date.available2025-01-20T22:03:06Z
dc.date.issued2021
dc.description.abstractWe define certain arithmetic derivatives on Z that respect the Leibniz rule, are additive for a chosen equation a + b = c, and satisfy a suitable nondegeneracy 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.
dc.fuente.origenWOS
dc.identifier.doi10.4153/S0008439521000990
dc.identifier.eissn1496-4287
dc.identifier.issn0008-4395
dc.identifier.urihttps://doi.org/10.4153/S0008439521000990
dc.identifier.urihttps://repositorio.uc.cl/handle/11534/94016
dc.identifier.wosidWOS:000741812800001
dc.language.isoen
dc.revistaCanadian mathematical bulletin-bulletin canadien de mathematiques
dc.rightsacceso restringido
dc.subjectArithmetic derivative
dc.subjectabc Conjecture
dc.subjectGeometry of Numbers
dc.titleArithmetic derivatives through geometry of numbers
dc.typeartículo
sipa.indexWOS
sipa.trazabilidadWOS;2025-01-12
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