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Linear dependence of a function set of m variables with vanishing generalized Wronskians. (English) Zbl 0724.15004
The author considers necessary and sufficient conditions for a set $\phi$ of n functions ${\phi }_{i}:{E}^{m}\to {E}^{1}$, which together with their partial derivatives of order at least n-1 are continuous, to be linearly dependent. After giving some definitions he shows that the vanishing of all generalized Wronskians of $\phi =\left({\phi }_{1}\left(t\right),···,{\phi }_{n}\left(t\right)\right)$, $\left(t=\left({t}_{1},···,{t}_{m}\right)\right)$ in an open set $G\subset {E}^{m}$ implies that G contains a countable set of disjoint, open, connected components of the interiors of set of constant order such that (1) on each such component $\phi$ is linearly independent, (2) the union of these components is dense in G.
##### MSC:
 15A03 Vector spaces, linear dependence, rank 53A45 Vector and tensor analysis 26B12 Calculus of vector functions