Treffer: Bezout identities with pseudopolynomial entries

If \(f_1,f_2, \dots, f_s\) are coprime polynomials in \(z\in\mathbb{C}\) then there exist polynomials \(q_1,q_2, \dots, q_s\) such that \[ 1= \sum^s_{i=1} f_i(z) \cdot q_i(z). \] This is the well known Bezout identity. The purpose of this paper is to establish an analogous identity in the case where \(f_1,f_2,\dots,f_s\) are entire functions in \(\mathbb{C}^n\) of the form \[ f(z)= \sum^m_{k=0} a_k(z') \cdot z_1^{m-k} \] where \(m=m(f)\) is entire and \(a_k(z')\) are entire functions in \(z'=(z_2,z_3, \dots, z_n) \in \mathbb{C}^{n-1}\), satisfying some growth conditions.