Let $f, g \in \mathbb{Q}[x_1, ..., x_n]$ be polynomials of degree $d$. Let $F$ and $G$ denote the degree $d$ portions of $f$ and $g$ respectively.
Suppose there exists a non-singular point $\mathbf{x}_0 \in (0,1)^n$ of $f(\mathbf{x}) = g(\mathbf{x}) = 0$. Does it then follow that there exists a non-singular point $\mathbf{y}_0 \in (0,1)^n$ of $F(\mathbf{x}) = G(\mathbf{x}) = 0$?
I feel like it should but maybe not... I would appreciate any hint or explanation on how I can see this. Thank you very much!