2022/10/13 by Mondal, Golam Mostafa
#32E20 #41A10 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.07214
In this paper, we first consider the graph of (F1,F2,⋯,Fn) on \mathbbDn, where Fj(z)=z^mjj+Rj(z),j=1,2,⋯,n, which has non-isolated CR-singularities if mj>1 for some j∈\1,2,⋯,n\. We show that under certain condition on Rj, the graph is polynomially convex and holomorphic polynomials on the graph approximates all continuous functions. We also show that there exists an open polydisc D centred at the origin such that the set \(z^m11,⋯, z^mnn, z1^mn+1 + R1(z),⋯, zn^m2n + Rn(z)):z∈ D,mj∈ ℕ, j=1,⋯,2n\ is polynomially convex; and if gcd(mj,mk)=1~~∀ j\not=k, the algebra generated by the functions z^m11,⋯, z^mnn, z1^mn+1 + R1,⋯, zn^m2n + Rn is dense in C(D). We prove an analogue of Minsker's theorem over the closed unit polydisc, i.e, if gcd(mj,mk)=1~~∀ j\not=k, the algebra [z^m11,⋯, z^mnn, z1^mn+1,⋯ , zn^m2n;\mathbbDn ]=C(\mathbbDn). In the process of proving the above results, we also studied the polynomial convexity and approximation of certain graphs.