Integral closure algorithm

Let $A$ be a ring, $I \subset A$ an ideal. We propose an algorithm to compute $\overline{I} = \bigl\{b \in A \:\big\vert\: b$ is integral over $I\bigr\}$, the integral closure of $I$. $I$ is called integrally closed if and only if $I = \overline{I}$. It is called normal if $I^k = \overline{I^k}$ for all $k>0$. Note that $I \subset \overline{I} \subset \sqrt{I}$. We are mainly interested in the case $A = K[x_1, \dots, x_n]$.

In the following, we describe an algorithm to compute $\overline{I^k}$ for all $k$, simultaneously. Consider the Rees algebra $\mathcal{R}(I) = \bigoplus_{k \ge 0} I^k t^k \subset A[t]$, and let $\widetilde{\mathcal{R}(I)}$ denote the integral closure of $\mathcal{R}(I)$ in $A[t]$. Then

$$\widetilde{\mathcal{R}(I)} = \textstyle{\bigoplus\limits_{k \ge 0}} \overline{I^k} t^k \subset A[t]\,.$$

If $A$ is normal, then $A[t]$ is normal and hence, the normalization of $\mathcal{R}(I)$, that is, the integral closure of $\mathcal{R}(I)$ in $Q(\mathcal{R}(I))$, satisfies

$$\overline{\mathcal{R}(I)} = \widetilde{\mathcal{R}(I)} = \textstyle{\bigoplus\limits_{k \ge 0}} \overline{I^k} t^k\,.$$

Hence, computing the normalization of $\mathcal{R}(I)$ provides the integral closure of $\overline{I^k}$ for all $k$.

To be specific, let $A = K[{\boldsymbol{x}}] = K[x_1, \dots, x_n]$, $\,I = \langle f_1, \dots,f_k\rangle \subset A$ with $K$ a perfect field. Then

$$\mathcal{R}(I)\, = \,K[{\boldsymbol{x}}, tf_1, \dots, tf_k] \, \xleftarrow[\varphi]{\cong}\, K[{\boldsymbol{x}}, U_1, \dots, U_k]\big/(Ker\varphi)$$

where $\varphi : K[{\boldsymbol{x}},{\boldsymbol{U}}] \to K[{\boldsymbol{x}},t]$ maps $x_i \mapsto x_i$, $\,U_j \mapsto t f_j$. $Ker\varphi$ can be computed by eliminating $t$ from

$$J := \langle U_1 - tf_1, \dots, U_k - tf_k \rangle \subset K[{\boldsymbol{x}},{\boldsymbol{U}},t]\,,$$

that is, $Ker\varphi = J \cap K[{\boldsymbol{x}},{\boldsymbol{U}}]$. For the integral closure of $I$ we need to compute

$$\UseComputerModernTips\xymatrix @C=0pt@R=2pt{ & \mathcal{R}(I) &\... ...upset &\overline{\mathcal{R}(I)} & =\, K[T_1, \dots, T_s]/I'\,. \ar[uuurrr] }$$

This means that we compute $\overline{\mathcal{R}(I)}$ as an affine ring $K[\boldsymbol{T}]/I'$ and, in each inductive step during the computation of $\overline{\mathcal{R}(I)}$, we also compute the map from the intermediate ring to $K[{\boldsymbol{x}},t]$.

The algorithm then reads as follows:

Input

$f_1, \dots, f_\ell \in K[x_1, \dots, x_n]$,$k\geq 1$ an integer, $I:=\langle f_1, \dots, f_\ell\rangle$.

Output

Generators for $\overline{I^k} \subset K[x_1, \dots, x_n]$.

1. Compute the Rees algebra $\mathcal{R}(I) \subset K[{\boldsymbol{x}},t]$.
2. Compute the normalization $\overline{\mathcal{R}(I)}$, together with maps $\varphi, \psi$, so that
   $$\UseComputerModernTips\xymatrix @C=1pt@R=6pt{ & \mathcal{R}(I) \a... ...mathcal{R}(I)} \ar@{=}[rrr] &&& K[T_1, \dots, T_s]/J\ar@{^{(}->}[uu]_-{\psi} }$$
   commutes.
3. Determine $a_i, b_i \in \mathcal{R}(I)$, so that $T_i = \tfrac{a_i}{b_i} \in Q\bigl(\mathcal{R}(I)\bigr)$ compute $\tfrac{\varphi(a_i)}{\varphi(b_i)}\in K[{\boldsymbol{x}},t]$ (indeed, we find a universal denominator $b=b_i$ for all $i$).
4. Determine generators $g_1,\dots,g_s$ of the $K[\boldsymbol{x}]$-ideal which is mapped to the component of $t$-degree $k$ of the subalgebra $\psi\bigl(\overline{\mathcal{R}(I)}\bigr) \subset K[{\boldsymbol{x}},t]$.
5. Return $g_1,\dots,g_s$.

The algorithms described above are implemented in SINGULAR and contained in the libraries normal.lib [16] and reesclos.lib [23] contained in the distribution of SINGULAR 2.0 [17]. Similar procedures can be used to compute the conductor ideal of $A$ in $\overline{A}$. An implementation will be available soon.