When is a homogeneous polynomial an inner function on the unit torus?

Question

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial is equal to 1 almost every where on distinguished boundary of the torus.Any known criteria?

Answer 1

This happens only for monomials $z^m w^n$. Indeed, write $p(z,w)=\sum c_k z^k w^{n-k}$. For each $|\lambda|=1$ and $|w|=1$, we then have $|p(\lambda w, w)|=|\sum_k c_k \lambda^k|=1$, which means that $q(\lambda)=\sum c_k\lambda^k$ is an inner function on the unit circle, and thus $q$ is a monomial.