Note on bases in algebras of analytic functions on Banach spaces

Let $\{P_n\}_{n=0}^\infty$ be a sequenceof continuous algebraically independent  homogeneous polynomials on a complex Banach space $X.$ We consider the following question: Under which conditions polynomials $\{P_1^{k_1}\cdots P_n^{k_n}\}$ form a Schauder (perhaps absolute) basis in the minimal subalgebra of entire functions of bounded type on $X$ which contains the sequence $\{P_n\}_{n=0}^\infty$? In the paper we study the following examples: when $P_n$ are coordinate functionals on $c_0,$ and when $P_n$ are symmetric polynomials on $\ell_1$ and on $L_\infty[0,1].$ We can see that for some cases $\{P_1^{k_1}\cdots P_n^{k_n}\}$ is a Schauder basis which is not absolute but for some cases it is absolute.