Pasten's arithmetic derivatives and the decomposition into gains

Source 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 $X=\log c/\log\operatorname{rad}(abc)$ 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 $c/\operatorname{rad}(abc)$ 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 $1+2400=2401$.