Pasten's arithmetic derivatives and the decomposition into gains
Abstract
Pasten attached to each coprime triple a+b=c a lattice of arithmetic derivations that obey the Leibniz rule and are additive on this one equation. An arithmetic Wronskian then gives an abc-type inequality governed by the norm |ψ| of the derivation, and Pasten showed that sufficiently small derivations exist under the abc conjecture and give an abc inequality with a fixed exponent. We determine exactly which values this Wronskian takes, and we show that its estimate with exact prime weights has the best possible coefficient. The quality X=log c/lograd(abc) factors into an approximation gain and a power gain for each chosen presentation of the triple, and the resulting inequality bounds X and both gains. For coprime polynomials the same argument gives Ga < 1 and Gp < 3. We also show that the largest absolute value of the derivation on a, b and c exceeds c/rad(abc) whenever the derivation is additive on the triple with non-zero Wronskian, so that it gives no bound on the quality by itself, and we determine its minimum when a=1. For Fermat triples the presentation by bases has approximation gain greater than n/3, which recovers Pasten's conditional asymptotic Fermat theorem. Finally, for five triples we certify the minimal norms and find the inequality within $0.08$ of the quality, with equality for the triples 1+80=81 and 1+2400=2401.
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BibTeX
@misc{laniewski2026pastens,
title = {{Pasten's arithmetic derivatives and the decomposition into gains}},
author = {R. Laniewski and K. Müller},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26490},
eprint = {2610.26490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26490},
}
RIS
TY - UNPB
TI - Pasten's arithmetic derivatives and the decomposition into gains
AU - R. Laniewski
AU - K. Müller
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26490
UR - https://arxiv.org/abs/2610.26490
AB - Pasten attached to each coprime triple a+b=c a lattice of arithmetic derivations that obey the Leibniz rule and are additive on this one equation. An arithmetic Wronskian then gives an abc-type inequality governed by the norm |ψ| of the derivation, and Pasten showed that sufficiently small derivations exist under the abc conjecture and give an abc inequality with a fixed exponent. We determine exactly which values this Wronskian takes, and we show that its estimate with exact prime weights has the best possible coefficient. The quality X=log c/lograd(abc) factors into an approximation gain and a power gain for each chosen presentation of the triple, and the resulting inequality bounds X and both gains. For coprime polynomials the same argument gives G_a < 1 and G_p < 3. We also show that the largest absolute value of the derivation on a, b and c exceeds c/rad(abc) whenever the derivation is additive on the triple with non-zero Wronskian, so that it gives no bound on the quality by itself, and we determine its minimum when a=1. For Fermat triples the presentation by bases has approximation gain greater than n/3, which recovers Pasten's conditional asymptotic Fermat theorem. Finally, for five triples we certify the minimal norms and find the inequality within $0.08$ of the quality, with equality for the triples 1+80=81 and 1+2400=2401.
ER -