  In this paper we present a constructive method to characterize ideals of the
local ring $\mathscr{O}_{\mathbb{C}^n,0}$ of germs of holomorphic functions at
$0\in\mathbb{C}^n$ which arise as the moduli ideal $\langle f,\mathfrak{m}\,
j(f)\rangle$, for some $f\in\mathfrak{m}\subset\mathscr{O}_{\mathbb{C}^n,0}$. A
consequence of our characterization is an effective solution to a problem
dating back to the 1980's, called the Reconstruction Problem of the
hypersurface singularity from its moduli algebra. Our results work regardless
of whether the hypersurface singularity is isolated or not.

Algebraic Geometry