I have seen being mentioned that algebraic independence of polynomials can be tested by the so called Jacobian Criterion (Apparently one takes the Jacobian matrix of these polynomials and inspects the rank of the matrix (or the rank of its minors)). Where can i find the precise statement and its proof?
For a combinatorial (!!!) proof see Theorem 2.2 from this paper.
Another reference seems to be S. Lefschetz, Algebraic Geometry, 1953, Ch. I, 11.4.