The following is a theorem from the book Ideals, Varieties and Algorithms:
Let k be an algebraically closed field and let I be a homogeneous ideal in $k[x_0, \dots , x_n]$. If $\dim V(I) > 0$ and $f$ is any nonconstant homogeneous polynomial, then $\dim V(I) \geq \dim V(I + (f)) \geq \dim V(I ) − 1$. This implies that if $f_1, \dots , f_r$ are nonconstant homogeneous polynomials in $k[x_0, \dots , x_n]$, then $\dim V(f_1, \dots, f_r) \geq n − r$.
Then it's stated in the book that the above may not hold for affine varieties and a counterexample is:
$$\dim V(xy, yz, z-1) = 0 = \dim V(xy, yz) -2.$$
They also mention that it's possible to formulate a version of the above theorem for affine varieties, but that's beyond the scope of the book. That's exactly what I want to know.
To be precise, what is a sufficient condition under which $\dim V(f_1,\dots, f_r) \geq n − r$ when $f_1,...,f_r$ are nonconstant (possibly non-homogeneous) polynomials in $k[x_1,...,x_n]$. I suspect that a sufficient condition might be that the constant terms in $f_1,...,f_r$ are all $0$s. Is that true?