On a criterion for algebraic independence and its variants

From around 2010 onward, Elsner et al.,developed and applied a method in which the algebraic independence of n quantities x_1,…,x_n over a field is transferred to further n quantities y_1,…,y_n by means of a system of polynomials in 2n variables X_1,…,X_n,Y_1,…,Y_n. In this paper, we systematically study and explain this criterion and its variants.