- ,
- ,
- .

(12) |

is a -adic approximation of at order . By Proposition 4 there exists a polynomial such that

(13) |

Since we have

we deduce from Proposition 7

(14) |

Since this shows that is in the ideal generated by . Similarly, is in the ideal generated by . Therefore we can divide and by , leading to

(15) |

Now observe that

(16) |

Let us denote by the canonical homomorphism from to . Then we obtain

Now, since holds we have

(18) |

Since holds we can solve Equatiion 17 for . Finally, from Theorem 1, we have such that we can

(19) |

- for every we have .
- the Jacobian matrix is left-invertible modulo .

- for every we have ,
- for every we have .

(20) |

and

(21) |

Since is finitely generated, then so is and let such that

(22) |

Therefore, for every , there exist such that

(23) |

For each we want to compute such that

(24) |

is the desired

(25) |

Using Proposition 5 we obtain

(26) |

where is the Jacobian matrix of at .

Hence, solving for such that

leads to solving the system of linear equations:

for and .

Now using for we obtain

(28) |

Therefore the linear system equations given by Relation (27) has solutions.

*
*

2008-01-07