Weierstrass's Nullstellensatz via the axiom of choice or how to construct holomorphic functions from highly discontinuous functions

Abstract

Based on the axiom of choice we revisit a method to prove in simply connected domains the existence of a holomorphic function with prescribed zeros.

Keywords:

• Weierstrass's Nullstellensatz
• axiom of choice
• meromorphic functions

AMS Subject Classifications:

• 30D10
• 30E99

Acknowledgments

We thank Peter Pflug for reference [6]. We learned from Robert B. Burckel that this approach (=meromorphic to holomorphic) of the Weierstrass theorem for simply connected domains is due to Mittag-Leffler and is mentioned as an exercise in his book [3, p. 391], the hints given there being more involved, though, and that the general version is in [2, p. 248]. We thank him for this information and for having caught some typos. Finally, we thank the referee(s) for their valuable two-page long report.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This step needs the axiom of choice: on the set of all polygonial arcs in $\tilde{U}$ which have $z_0$ as the initial point consider the equivalence relation ${\gamma }_1\sim {\gamma }_2$ if and only if both ${\gamma }_j$ have a common endpoint. Then from each equivalence class one chooses one member.

2 This is the only place where we need the assumption on the simply connectedness.

3 If we replace in (1(1) $\frac{1}{2\mathrm{\pi i}}{\int }_{\gamma }g\left(\xi \right)\mathrm{d}\xi =\sum \underset{=:n_j}{\underbrace{n(\gamma ,w_j)}}\mathrm{Res}(g,w_j)=\sum n_j{k}_j\in \mathbb{Z}.$(1) ) the closed path γ by the closed path ${\gamma }_z\oplus {\varphi }_z^{-1}$, where ${\varphi }_z$ is another path connecting $z_0$ with z, then we actually see that the function $e^G$ is independent of the path chosen that connects $z_0$ with z.

4 We think that this is also a magical behaviour since for any function m holomorphic in a domain Ω excepted at some isolated singularities ${\xi }_n$, the function $e^m$ has an essential singularity at ${\xi }_n$. As explained above, this ‘magical behaviour’ comes from the fact that G may not even by measurable.