The Strong Nullstellensatz for Certain Normed Algebras

TL;DR

This paper proves a version of the Nullstellensatz for certain closed ideals in the Banach algebra of absolutely convergent power series on the polydisc, extending classical algebraic geometry results to a functional analytic setting.

Contribution

It establishes a strong Nullstellensatz for ideals closed in the $ ext{l}_1$-norm completion of polynomial rings, a result not holding in general Banach algebras.

Findings

Nullstellensatz holds for closed ideals in the $ ext{l}_1$-norm polynomial algebra.

Describes the closure of ideals in specific cases.

Shows the partial extension of algebraic geometry principles to Banach algebras.

Abstract

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.