A set of polynomials $\{f_1,\ldots,f_m\}$ in $k[x_1,\ldots,x_n]$ are algebraically independent over $k$ iff for all polynomials $p \in k[y_1,\ldots,y_m]$, $p(f_1,\ldots,f_m) = 0$ implies that $p = 0$.
In linear algebra, the dimension of a subspace of $k^n$ defined by $m$ linearly independent equations is $n - m$. Is the analogous statement in algebraic geometry true: that the dimension of a variety in $k^n$ defined by $m$ algebraically independent polynomials is $n - m$?
Thanks!