Homogeneous polynomials on the ball of $\textbf {C}^ 2$

Abstract:

Let ${B_2}$ be the open unit ball of ${C^2}$. Let $T = \left \{ {\left ( {u,w} \right ) \in b{B_2}:1 - a \leq {{\left | u \right |}^2} \leq a} \right .$ where $1/2 < a < 1/2 + \sqrt 3 /4$. For each $n \in {\mathbf {N}}$ we construct a homogeneous polynomial ${H_n}:{C^2} \to {C^2}$ of degree $n$ such that $\left | {{H_n}} \right | \leq 1$ on ${B_2}$ and $\left | {{H_n}} \right | \geq c$ on $T$ where $c > 0$ does not depend on $n$.