Ideal growth in amalgamated powers of nilpotent rings of class two and zeta functions of quiver representations

Let (Formula presented.) be a nilpotent algebra of class two over a compact discrete valuation ring (Formula presented.) of characteristic zero or of sufficiently large positive characteristic. Let (Formula presented.) be the residue cardinality of (Formula presented.). The ideal zeta function of (Formula presented.) is a Dirichlet series enumerating finite-index ideals of (Formula presented.). We prove that there is a rational function in (Formula presented.), (Formula presented.), (Formula presented.), and (Formula presented.) giving the ideal zeta function of the amalgamation of (Formula presented.) copies of (Formula presented.) over the derived subring, for every (Formula presented.), up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver representations of class two defined by Lee and Voll, and in particular for Dirichlet series counting graded submodules of a graded (Formula presented.) -module. If the algebra (Formula presented.), or the quiver representation, is defined over (Formula presented.), then we obtain a uniform rationality result.