Let X be a projective variety. In this talk, we explain some of the difficulties of studying Bir(X) in comparison with Bir($\mathbb{P}^n$). We define the concept of birational maps of clear polynomial degree d over an arbitrary projective variety. We show how to replace classical techniques such as the Jacobian criterion with commutative algebraic counterparts such as analytic spread and Hilbert functions that provide facilities to study Bir(X) in full generality. We show that the loci of ideals in the principal class, ideals of grade at least two, and ideals of maximal analytic spread are Zariski open sets in the parameter space. As an application, we show that the set of birational maps of clear polynomial degree d over an arbitrary projective variety X, denoted by Bir(X)$_d$, is a constructible set. This extends a previous result by Blanc and Furter.