---
abstract: 'We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic $4$-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\frac{b}{a} \in {\mathbb{N}}$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the ECH capacities give sharp inequalities for embedding ellipsoids into polydisks.'
address:
- 'Department of Mathematical Sciences, IUPUI, Indianapolis, USA.'
- 'Department of Mathematics, The University of Western Ontario, London, Canada.'
author:
- Olguta Buse
- Martin Pinsonnault
title: 'Packing numbers of rational ruled $4$-manifolds'
---

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Introduction and main results
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Background
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Let $\sqcup_k B(c)$ be the disjoint union of $k$ standard $2n$-dimensional balls of radius $r$ and capacity $c=\pi r^{2}$. The $k^{\text{th}}$ packing number of a compact, $2n$-dimensional, symplectic manifold $(M, \omega)$ is $$p_k (M,\omega) = \frac{\sup_c \operatorname{vol}(\sqcup_k B(c)}{\operatorname{vol}(M,\omega)}$$ where the supremum is taken over all $c$ for which there exists a symplectic embedding of $\sqcup_k B(c)$ into $(M,\omega)$. Naturally, $p_k(M,\omega) \leq 1$. When 