If $P$ is a polynomial in ${\mathbf{R}}^n$ such that $1/P$ integrable, then the inverse Fourier transform of $1/P$ is a fundamental solution $E_P$ to the differential operator $P\left(D\right)$. The purpose of the article is to study the dependence of this fundamental solution on the polynomial $P$. For $n=1$ it is shown that $E_P$ can be analytically continued to a Riemann space over the set of all polynomials of the same degree as $P$. The singularities of this extension are studied.