# Flatness through parametrization

Let $$A$$ be a $$\mathbb{C}$$-algebra. Let $$\phi:\mathbb{C}[X_1,...,X_n] \otimes_{\mathbb{C}} A \to \mathbb{C}[t] \otimes_{\mathbb{C}} A$$ be a ring homomorphism, sending $$X_i$$ to say $$f_i \in \mathbb{C}[t] \otimes_{\mathbb{C}} A$$. I am looking for conditions on $$f_i$$ such that $$A[X_1,...,X_n]/(\mbox{ker}(\phi))$$ is $$A$$-flat. Any reference will be most welcome.