Factorization of polynomials with estimates of norms

We want to factorize the polynomials with the estimate of norms and degrees of factors. This problem has connections with many different questions in Functional and Harmonic analysis including the Fejer-Riesz factorization, Peller's lemma on trace norm of finite Hankel matrix, some extension problems for finite Toeplitz matrices, and a recent result of Ferguson and Lacey on the weak factorization of analytic functions in the bidisc. We try to establish elementary relations between these questions. We give a certain elementary factorization theorem for polynomials of two variables.