Holomorphic automorphic forms on irreducible bounded symmetric domains are a natural and far-reaching generalization of classical modular forms and Siegel modular forms. Every irreducible bounded symmetric domain \(D\) can be realized as a quotient \(G/K\), where \(G\) is a connected semisimple Lie group with finite center, and \(K\) is a maximal compact subgroup of \(G\). In this talk, we will consider holomorphic automorphic forms on \(D\) given by Poincaré series of polynomial type that lift, in a standard way, to cuspidal automorphic forms on \(G\) given by Poincaré series of \(K\)-finite matrix coefficients of integrable discrete series representations of \(G\). Determining which of these Poincaré series vanish identically is a surprisingly subtle and non-trivial problem. I will explain how it can be answered, in certain cases, by means of Muic's integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups.