Let $f_1, f_2, \ldots, f_m$ be polynomials in $n$ variables, $n>m.$ Suppose we know that any $m-1$ of the polynomials are functionally independent. Can we deduce now that all $m$ polynomials are functionally independent?
Polynomials $f_1, f_2, \ldots, f_m$ in $n$ variables, $m \leq n$, are called functionally independent if the rank of its Jacobian matrix equals exactly $m.$