Given an "explicit" function $f:\mathbb{R}^n\to\mathbb{R}^n$, (e.g $F(x_1,\dots x_n)=\cos(x_n)+x_1^2e^{x_2}$) under some assumptions one can allegedly develop a Fourier transform given by $$F[f(x)](\xi)=\int_{\mathbb{R}^n}f(x_1,\dots,x_n)\exp(-2\pi i \langle x,\xi\rangle)dx$$
Now, given a polynomial implicit function $F:\mathbb{R}^n\to\mathbb{R}^n$ which is defined as zeroes of equation, i.e. $$F=\{\sum_{i=1}^m\prod_{j=1}^n x_{j,i}^{p_{j,i}}=0\}$$ can one develop Fourier expansion for it in respect to $(\xi_1\dots \xi_n)$? If one can, what do we require in order to make it well defined?