Rational function with absolute value $1$ on unit circle 
What is the general form of a rational function which has absolute value $1$ on the circle $|z|=1$? In particular, how are the zeros and poles related to each other?

So, write $R(z)=\dfrac{P(z)}{Q(z)}$, where $P,Q$ are polynomials in $z$. The condition specifies that $|R(z)|=1$ for all $z$ such that $|z|=1$. In other words, $|P(z)|=|Q(z)|$ for all $z$ such that $|z|=1$. What can we say about $P$ and $Q$?

Answer your question about why $M$ is constant: it's simply because $M$ is a quotient of two polynomials. If the quotient is $1$ on the unit circle, it means these two polynomials are equal at all the points of the circle. This implies that these two polynomials are the same. So $M$ is identically $1$. 
