Let $R$ be a commutative noetherian domain, $S=R[x_1,\ldots,x_n]$ and $S_1=S/(f)$ , $f\in S$. Then $S_1$ is flat over $R$ if for every maximal ideal $\mathfrak m$ in $R$ the ring $S_1 \otimes_R R/\mathfrak m$ is not equal to $S \otimes_R R/\mathfrak m$.
This can be found in Milne, Etale Cohomology, Chapter 1, Remark 2.6(a); I have seen this nice result thanks to this answer.
My question:
Can this theorem be somehow generalized to the case where $S_1=S/(f_1,\ldots,f_m)$, $f_1,\ldots,f_m \in S$? Is it easier to answer for $n=1$ or $n=2$?
(Maybe if $n > m$ there is an answer; maybe in this case $S_1$ is never flat over $R$.)