A type of generalized Wronskian is used for the induction step in the proof of Roth's theorem in Diophantine approximation. There's a nice exposition in Schmidt's lecture note, for example, but undoubtedly it can be found in lots of other places. Roughly, the way that generalized Wronskians are used in the proof of Roth's theorem is to  start with a polynomial $f(x_1,\ldots,x_r)$ and to construct a generalized Wronskian $W(f)$ that is the determinant of a matrix of partial derivatives (with controlled order) of $f$ having the property that $W(f)$ factors as
$$ W\bigl(f(x_1,x_2,\ldots,x_r)\bigr) = g(x_1)h(x_2,\ldots,h_r). $$
This factorization is the key to doing an induction on the number of variables in the auxiliary polynomial. 

<cite authors="Schmidt, Wolfgang M.">_Schmidt, Wolfgang M._, Diophantine approximation, Lecture Notes in Mathematics. 785. Berlin-Heidelberg-New York: Springer-Verlag. 299 p.(1980). [ZBL0421.10019](https://zbmath.org/?q=an:0421.10019).</cite>