If it is self-referential, it indeed is. The parent comment (this comment's grandparent) is not noise. As with almost any question involving the term "random", the answer depends on what you mean by that term.

Oftentimes, one can leave that implicit because there is a commonly understood distribution. For example, "pick a number between 1 and n" typically implicitly assumes a discrete uniform distribution, "pick a number between 0 and 1" a continuous uniform distribution.

For polynomials of degree n, whether picking a random polynomial by randomly selecting its coefficients is more obvious than doing zo by randomly selecting its zeroes depends on what you pick as the natural representation of a polynomial.