Let $F_1(\mathbf{x}), F_2(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ be a degree $d$ homogeneous polynomial. For the system of equations $$F_1(\mathbf{x})= F_2(\mathbf{x}) =0,$$ we have the following singular integral $$ \sigma_{\infty} = \int_{\mathbb{R}^2} I(\alpha_1, \alpha_2) d\alpha_1d\alpha_2, $$ where $$ I(\alpha_1, \alpha_2) = \int_{[0,1]^n} e^{2 \pi i (F_1(\mathbf{x}) \alpha_1 + F_2(\mathbf{x}) \alpha_2 ) } dx_1...dx_n. $$

Let $G_1(\mathbf{x}) = F_1(\mathbf{x}) + g(\mathbf{x})F_2(\mathbf{x})$ and $G_2(\mathbf{x}) = F_2(\mathbf{x})$, where $g$ is some homogeneous polynomial. Then the set of solutions of $$G_1(\mathbf{x})= G_2(\mathbf{x}) =0$$ is the same as that of $F_1$ and $F_2$. Let $\sigma'_{\infty}$ be the singular integral of this system obtained by replacing $F_1$ and $F_2$ with $G_1$ and $G_2$, respectively, in the definition of singular integral above.

I suspect that $\sigma_{\infty}$ and $\sigma'_{\infty}$ are the same, since they both correspond to the same affine variety, but I don't see how I can prove this statement. Could anyone please give me a hint or explanation on how I can see this (assuming the integrals exists, etc)? Thank you very much.