In \({\mathbb{C}}^ 2\) let \(U^ 2=\{(z_ 1,z_ 2)|\) \(| z_ 1| <1\), \(| z_ 2| <1\}\), \(T^ 2=\{(z_ 1,z_ 2)|\) \(| z_ 1| =| z_ 2| =1\}\), and \(\bar U^ 2=\{(z_ 1,z_ 2)| | z_ 1| \leq | 1,\quad | z_ 2| \leq 1\}.\) A polynomial \(f(z_ 1,z_ 2)\) is said to be stable if \(f(z_ 1,z_ 2)\neq 0\) in \(\bar U^ 2\). For any fixed positive integers k,\(\ell,m\) and n we say that (k,\(\ell)\leq (m,n)\) if \(k\leq m\) and \(\ell \leq n\). If \(f(z_ 1,z_ 2)=\sum^{n}_{i=0}\sum^{m}_{i=0}a_{i_ j}z^ i_ 1z^ j_ 2,\) then (m,n) is said to be the degree of \(f(z_ 1,z_ 2).\)
Let \(A=\{f(z_ 1,z_ 2)| \deg f\leq (m,n)\}\), \(B=\{f(z_ 1,z_ 2)| f\in A\) and \(f(z_ 1,z_ 2)\neq 0\) in \(\bar U^ 2\}\), \(C=\{f(z_ 1,z_ 2)| f\in A\), \(f(z_ 1,z_ 2)\) has zeros in \(\bar U^ 2\}\); \(D=\{f(z_ 1,z_ 2)| f(z_ 1,z_ 2)\in A\), \(f(z_ 1,z_ 2)\neq 0\) on \(T^ 2\}\); \(E=\{f(z_ 1,z_ 2)| f(z_ 1,z_ 2)\in A\), \(f(z_ 1,z_ 2)\) has zeros on \(T^ 2\}\); \(F=\{f(z_ 1,z_ 2)| f(z_ 1,z_ 2)\in A\), \(f(z_ 1,z_ 2)\neq 0\) in \(\bar U^ 2\}\) and \(G=C\cap F\). It is easy to show that \(B\subset D\) and \(G\subset E.\)
The authors show that D and E are dense in A.