I'm interested in polynomials in several variables $p(x_1,\ldots,x_n)$, with complex coefficients, such that the maximum modulus of $p$ on the unit complex $n$-ball $ \max \{ |p(z_1,\ldots,z_n)| : \left|z_1\right|^2 + \cdots + \left|z_n\right|^2 \leq 1\} $ is known exactly. I can think of some, but these are mostly constructed from a 1-variable polynomial in the manner of $p(x,y) = q((x+y)/2)$ or such like. I'm particularly interested in the case $n = 2,3,4$ and polynomials of order less than a hundred or so.
Any takers?
Thanks in advance!