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checklist.md

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+# Things that we can do
+
+Given $f := (\tilde f,h) :  (\mathbb{C}^n,0) \to (\mathbb{C}^p\times \mathbb{C}^q,0)$ such that 
+$\tilde f : (\mathbb{C}^n,0) \to (\mathbb{C}^p,0)$ is finite.
+
+1. Set $X_i = f_i(x_1,\ldots,x_n)$ for $i = 1,\ldots, p$, and $Y_i = h_(x)$ for $i=1,\ldots, q$.  **done**
+1. Find a $\mathbb{C}$-basis $g_1,\ldots, g_m$ of $\mathcal{O}[x_1,\ldots,x_n]/(f_1,\ldots,f_p)$. **done**
+1. Given $\alpha \in \mathbb{C}[x_1,\ldots,x_n]^{p + q}$ such that there exists a unit $u\in \mathcal{O}[x_1,\ldots,x_n]$, and $A^1_1\ldots,A^{p+q}_m \in \mathbb{C}[X]$ with $u\cdot \alpha = g_1\cdot (A_1^1,\ldots, A_1^{p+q})+\cdots+g_m\cdot (A_m^1,\ldots, A_m^{p+q})$ modulo some submodule, find the $A_i^j$ and tell us if $u$ lives in $\mathcal{O}[X]$ or not.
+    - If there exists a solution with $u\in \mathbb{C}[x]$ **can be done with syzygies**.
+    - If there is a solution but not with $u$ being a polynomial, **to-do**.
+    - Even if in theory we are interested only in vectors of length $p+q$, in some cases it will be usefull to do it for length $1$.
+1. Apply the previous step to the vectors $(\frac{\partial f_1}{\partial x_j},\ldots, \frac{\partial f_{p+q}}{\partial x_j})$ to obtain $n$ vectors of length $m\times (p+q)$. Call them $(F_1^1,\ldots, F_{m\times (p+q)}^1), \ldots, (F_1^n,\ldots, F_{m\times (p+q)}^n)$.
+1. For each $(e_1,\ldots,e_q)$ with $0\leq e_i<m$, apply algorithm 3 to $h_1^{e_1}\cdots h_q^{e_q}$ to get vectors $H_1,\ldots, H_{(p+q)m^q}$. If in some point algorithm 3 said that it used a unit not in $\mathcal{O}[X]$, raise a nonimplemented error (we don't know how to handle this case).
+1. Compute a standard basis of the module $M$ generated by the $F$'s and the $H$'s.
+1. Use the standard basis to compute a $\mathbb{C}$-basis of the quotient $\mathcal{O}[X]^{m(p+q)}/M$. Its size is the Ae-codimension.
+
+
+
+
+# Pending generalizations:
+
+- Change the source by a Cohen-Macaulay subvariety.
+
+# Theoretical remarks:
+
+- The reason for taking $e_i<m$ in step 5 is because $Y_i^m$ can be expressed as a polynomial in $X,Y$ with degree in the $Y$ smaller than $m$. **check the theory**