Functional Dependence

Abstract

The condition for dependence of n functions of n+p variables is roughly that every determinant of order n formed from the matrix of the first partial derivatives vanish identically.The theorem easiest to prove assumes condition (A): One of the determinants of highest order which do not vanish identically is different from zero at a given point.The first theorem free from condition (A) is due to Bliss,f who established an analytic relation in the case of two analytic functions of not more than two variables.Osgoodf proved that, for the case of three or more analytic functions of as many variables, the identical vanishing of the Jacobian does not necessarily imply that the functions satisfy an analytic relation.No result of a positive nature was given in this case.More recently, Knopp and Schmidt have established a relation for the case of n real functions, of class C',\\ of not more than n variables.The result is obtained in the large, and free of condition (A).^fIn the present paper we treat first the real case, extending the results