Local polynomial convexity of graphs of functions in several variables

In this paper, we investigate the locally polynomial convexity of graphs of smooth functions in several variables. We also give a sufficient condition for a real analytic function g   defined near 0 in CC which behaves like z‾n near the origin so that the algebra generated by zmzm and g is dense in the space of continuous functions on D for all disks D   close enough to the origin in CC.