Generalized Wronskians
For n functions of several variables, a generalized Wronskian is the determinant of an n by n matrix with entries Di(fj) (with 0≤i<n), where each Di is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish. As in the 1 variable case the converse is not true in general: if all generalized Wronskians vanish this does not imply that the functions are linearly dependent. However the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see Wolsson (1989b).
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