On Tjurina Ideals of Hypersurface Singularities

The Tjurina ideal of a germ of an holomorphic function $f$ is the ideal of $\mathscr{O}_{\mathbbm{C}^n,0}$ - the ring of those germs at $0\in\mathbbm{C}^n$ - generated by $f$ itself and by its partial derivatives. Here it is denoted by $T(f)$. The ideal $T(f)$ gives the structure of closed subscheme of $(\mathbbm{C}^n,0)$ to the hypersurface singularity defined by $f$, being an object of central interest in Singularity Theory. In this note we introduce \emph{$T$-fullness} and \emph{$T$-dependence}, two easily verifiable properties for arbitrary ideals of germs of holomorphic functions. These two properties allow us to give necessary and sufficient conditions on an ideal $I\subset \mathscr{O}_{\mathbbm{C}^n,0}$, for the equation $I=T(f)$ to admit a solution $f$.
Comment: 18 pages. Included information about Reconstruction Problem and Reconstruction problem as well as linked the interest of the results obtained to the solution of both problems. In this process, different references were included as well. Comments are still welcome