Computing the non-normal locus

Input

$f_1, \dots, f_k \in S=K[x_1, \dots, x_n]$, $\;I := \langle f_1, \dots, f_k\rangle$.
We assume that $I$ is a radical ideal.

Output

Generators for $I_{\text{\it NN}}$ such that $V(I_{\text{\it NN}}) = NN(S/I)$.

1. Compute an ideal $I_{Sing}$ as described in Sect. .
2. Compute a non-zerodivisor $u \in I_{Sing}$: choose an $S$-linear combination $u$ of the generators of $I_{Sing}$ and test if $(I : u) := \bigl\{g \in S \:\big\vert\: gu \in I\bigr\}$ is zero, by using that $u$ is a non-zerodivisor iff $(I : u) = 0$.

   A sufficiently general linear combination gives a non-zerodivisor.

3. Compute a test ideal $J$, e.g., $J=\sqrt{\langle u,I\rangle}$ or $J=\sqrt{I_{Sing}}$.
4. Compute generators $g_1, \dots, g_\ell$ for $\langle u,I\rangle\!\!\::\!\!\:\bigl((u J\!\!\:+\!\!\:I)\!\!\::\!\!\:J\bigr)$ as $S$-module.
5. Return $\{g_1, \dots, g_\ell\}$.