The quality plane of $abc$-triples and the decomposition into gains

Abstract

To a coprime triple $a+b=c$ we attach two coordinates, the quality $X$ and the logarithmic mass $Y$, whose ratio $λ$ measures the balance of the triple and lies asymptotically between $2$ and $3$. In these coordinates the $abc$ conjecture and Szpiro's conjecture for Frey curves become asymptotic bounds with thresholds $X=1$ and $Y=3$, and the direct implications between them follow from the range of $λ$. We then place the approximation and power gains of Müller, Taktikos and de Weger in the same plane. Relative to a chosen presentation of the triple, one exponent factor $f$ relates the gains to the coordinates, and Szpiro's conjecture for this family is equivalent to the asymptotic bound $f\le 3$ uniformly over all presentations. Under the two separate asymptotic gain bounds for a fixed choice of presentations, we obtain $\limsup X\le 9/2$ and $\limsup Y\le 27/2$ as the radical tends to infinity. Stronger conclusions depend on the presentation or on a joint bound for the gains. We close with numerical data and two research objectives.