  Consider the polynomial ring in any finite number of variables over the
complex numbers, endowed with the $\ell_1$-norm on the system of coefficients.
Its completion is the Banach algebra of power series that converge absolutely
on the closed polydisc. Whereas the strong Hilbert Nullstellensatz does not
hold for Banach algebras in general, we show that it holds for ideals in the
polynomial ring that are closed for the indicated norm. Thus the corresponding
statement holds at least partially for the associated Banach algebra. We also
describe the closure of an ideal in small cases.
