Closed ideals with countable hull in algebras of analytic functions smooth up to the boundary.

We denote by $\bbt$ the unit circle and by $\bbd$ the unit disc. Let $\calb$ be a semi-simple unital commutative Banach algebra of functions holomorphic in $\bbd$ and continuous on $\overline{\bbd}$, endowed with the pointwise product. We assume that $\calb$ is continously imbedded in the disc algebra and satisfies the following conditions: \\ (H1) The space of polynomials is a dense subset of $\calb$. \\ (H2) $\lim_{n\to +\infty}\|z^n\|_{\calb}^{1/ n}=1$.\\ (H3) There exist $k \geq 0$ and $C > 0$ such that \begin{eqnarray*} \big| 1- |\lambda| \big|^{k} \big\| f \big\|_{\calb} \leq C \big\| (z-\lambda) f \big\|_{\calb}, \quad (f \in \calb, |\lambda| < 2) \end{eqnarray*} When $\calb$ satisfies in addition the analytic Ditkin condition, we give a complete characterisation of closed ideals $I$ of $\calb$ with countable hull $h(I)$, where $$ h(I) = \big\{ z \in \overline{\bbd} : \, f(z) = 0, \quad (f \in I) \big\}. $$ Then, we apply this result to many algebras for which the structure of all closed ideals is unknown. We consider, in particular, the weighted algebras $\ell^1(\omega$) and $L^1(\bbr^{+},\omega)$