Assume that I have $k$ polynomials $f_1(x_1,\ldots x_n),f_2(x_1,\ldots x_n),\ldots f_k(x_1,\ldots x_n)$ in $n>k$ variables. Is it possible to calculate, ,i.e., does there exist a fast algorithm, the dimension of the variety $Z(f_1,\ldots f_k)$?

Does there exist a good criterion to check if the dimension of $Z(f_1,\ldots f_k)$ is $n-k$ when all $f_i$ are quadratic polynomials?