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.

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
Find a $\mathbb{C}$-basis g_1,\ldots, g_m of \mathcal{O}[x_1,\ldots,x_n]/(f_1,\ldots,f_p). done
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} can be done with syzygies, or reduction modulo de right ideal.
- If there is a solution but not with u being a constant, 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.
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).
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).
Compute a standard basis of the module M generated by the $F$'s and the $H$'s.
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

Relevant examples:

If f(x)=x + x^3, there is no solution for (3) with polynomial unit.

Comments

At some point we thought that solving (3) when u is a polynomial could be done with Syzygies, but we found a problem with that approach.

To do

Check if Pfistetr's and Hiroshi's algorithms can handle examples that we can.

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