A problem about generalization of Bezout equation to entire functions

Let $f_1,f_2,\ldots, f_n$ be $n$ entire functions, and they don't have any common zero as a whole (not in pairs), then can we assert that there exist $n$ entire functions $g_1,g_2,\ldots,g_n$,such that $F=f_1g_1+f_2g_2+\cdots+f_ng_n$ is zero free? We know that if $f_1,\ldots,f_n$ are known to be polynomials, the conclusion follows from Bezout equation and induction. But things become complicated when infinite products get involved.

(The original formulation of this problem is: Each finitely generated ideal in the ring of entire functions must be principal, which evidently can be reduced to the problem above.)

Solution 1:

This was originally proved in

O. Helmer, "Divisibility properties of integral functions", Duke Math. J. 6(1940), 345-356.

Notice that the result holds for the ring of holomorphic functions over any open connected subset of $\mathbb C$. See the book "Classical topics in complex function theory" by R. Remmert (GTM 172) for details and history.

Solution 2:

A proof of the fact that the ring of holomorphic functions on a connected open subset of the complex plane is a Bezout domain can be found in $\S 5.3$ of my commutative algebra notes. The proof uses some standard theorems in complex analysis: Weierstrass Factorization and Mittag-Leffler. If I remember correctly, the discussion is taken from Rudin's Real and Complex Analysis.