effective computational tool that can be used in many other instances where one must deal with Diophantine approximation problems involving hyperbolic lattices.

The authors would like to thank Yael Karshon and Dusa McDuff for their interest in this work and for many useful comments and suggestions. Many thanks to Michael Hutchings for providing us with an early version of [@H2] and sharing with us his computations for the ECH capacities of polydiscs. We thank MSRI where part of this work was completed.

Embedding balls in symplectic $4$-dimensional rational manifolds
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Symplectic embeddings of balls and symplectic blow-ups
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Using the correspondence between ball embeddings and blow-ups, the problem of deciding whether a collection $\mathcal{B}=\sqcup_{i}B(\delta_{j})$ of $k$ disjoint balls of capacities $\delta_{j}$ embeds symplectically in $M_{\mu}^{i}$ reduces to the question of understanding the symplectic cone of the $k$-fold blow-up of $M_{\mu}^{i}$ (see, for instance, McDuff-Polterovich [@MP]). Because the $k$-fold blow-up of $M_{\mu}^{i}$ is diffeomorphic to ${\mathbb{CP}}^{2}$ blown-up $(k+1)$ times, this is in turn equivalent to understanding the symplectic cone of the rational surfaces ${\mathbb{CP}}^{2}{\#\,(k+1)\overline{{\mathbb{CP}}}\,\!^2}$, for $k\geq 1$.

Reduced classes and symplectic cones of rational surfaces
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Given $n\geq 1$, let us denote by $X_{n}:=(X_{n},{\omega}_{\lambda;\delta_{1},\ldots,\delta_{n}})$ the $n$-fold symplectic blow-up of $({\mathbb{CP}}^{2},{\omega}_{\lambda})$ at 