On a criterion for algebraic independence applied to continued fractions

From around 2010 onward, Elsner et al.  developed and applied a method in which the algebraic independence of $n$ quantities $x_1,\dots ,x_n$ over a field is transferred to further $n$ quantities $y_1,\dots ,y_n$ by means of a system of polynomials in $2n$ variables $X_1,\dots ,X_n,Y_1,\dots ,Y_n$. In this paper, we systematically study and explain this criterion and its variants. Moreover, we present new results concerning its application to periodic non-regular continued fractions, namely continued fractions with real numbers as partial quotients. We show that given a continued fraction of this type, this criterion can be applied to prove that not only are the convergents algebraically independent from each other, but they are also algebraically independent from the continued fraction.