The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being dumb or I don't have the right tools to understand it. Here it goes, given the ring $\mathbb{C}[x_1,\ldots ,x_n]$ and n functions $f_i(x_1,\ldots, x_n)$ (quasihomogeneous if it matters) such that the ring $\mathbb{C}[x_1,\ldots,x_n]/(f_i)$ is a finite dimensional local artin algebra concentrated at the origin.

Now fix an integer d and consider $\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots y_n]$ with the function $ \widetilde{f}(x_i,y_i)=(\sum f_i(x_1,\ldots x_n)y_i)+p(y_1,\ldots,y_n)$ for polynomials p only depending on y's whose lowest order terms have total degree at least d. Can we find a p such that this function $\widetilde{f}$ has only isolated singularities?

The answer is yes, for d=1 I believe. I am actually interested in knowing if for any collection of f's as above there is some d>1 for which we can solve the problem. If this turns out to be impossible, a nice clean condition where we can find such a p would be great too. This problem seems open to brute force attacks, geometric interpretations, algebraic thinking (like maybe Grobner bases ?) so I am hoping someone will be able to know exactly what is going on. Thanks.