  Given $\u$ a multiplicative sequence of polynomial ideals, we consider the
associated algebra of holomorphic functions of bounded type, $H_{b\u}(E)$. We
prove that, under very natural conditions verified by many usual classes of
polynomials, the spectrum $M_{b\u}(E)$ of this algebra "behaves" like the
classical case of $M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded
type holomorphic functions). More precisely, we prove that $M_{b\u}(E)$ can be
endowed with a structure of Riemann domain over $E"$ and that the extension of
each $f\in H_{b\u}(E)$ to the spectrum is an $\u$-holomorphic function of
bounded type in each connected component. We also prove a Banach-Stone type
theorem for these algebras.
