The multiplicative Cousin problem and a zero set for the Nevanlinna class in the polydisc

Abstract

Let Qi be a polydomain in C n, the Nevanlinna class N(Q2) consists of all holomorphic functions f in fI such that log+If I has an n-harmonic majorant in Qi. Let Un be the open unit polydisc {z E Cn: tzl I 2. If there exists a constant 0 r, * , zn I > r and f (zl, ***Zn) = 0, then f has the same zeros as some function F E N(U ). In the above if limX.i 77(x) < 1, then Z(f) is a Rudin variety in which case there is a bounded holomorphic function with the same zeros as f.