Let $A=k[x_{1},...,x_{n}]$ be a polynomial ring over a field $k$, let $f_{1},...,f_{m} \in A$ be polynomials, and let $I=(f_{1},...,f_{m})$ be the ideal that they generate in $A$. Now suppose that we modify the $f_{i}$ to $\tilde{f}_{i}$, where $f_{i}=\tilde{f}_{i} \in A/I^{2}$.
I'd like to know when I can find a Zariski open set in $\mathbb{A}^{n}$ where $(f_{1},...,f_{m})=(\tilde{f}_{1},...,\tilde{f}_{m})$.
Here is an easy example when we can do this: suppose we have just a single polynomial $f$ and $\tilde{f}=f+a_{2}f^{2}+ \cdots +a_{k}f^{k}=f(1+a_{2}f+\cdots a_{k}f^{k-1})$. Then $(f)=(\tilde{f})$ where $(1+a_{2}f+ \cdots + a_{k}f^{k-1})$ doesn't vanish.
