Proof.
Let
![$ (a_0, a_1, \ldots, a_d)$](img31.png)
be a
![$ p$](img5.png)
-adic expansion of
![$ a$](img4.png)
w.r.t.
![$ p$](img5.png)
.
Let
![$ k < d + 1$](img121.png)
be a positive integer.
By Proposition
6,
the element
![$\displaystyle a^{(k)} = a_0 + a_1 p + \cdots a_{k-1} p^{k-1}$](img109.png) |
(12) |
is a
![$ p$](img5.png)
-adic approximation of
![$ a$](img4.png)
at order
![$ k$](img24.png)
.
By Proposition
4
there exists a polynomial
![$ {\psi} \in R[y]$](img83.png)
such that
![$\displaystyle {\phi}(a^{(k)} + a_k p^k) = {\phi}(a^{(k)}) + {\phi}'(a^{(k)}) a_k p^k + {\psi}(a^{(k)} + a_k p^k) (a_k p^k)^2.$](img122.png) |
(13) |
Since we have
we deduce from Proposition
7
![$\displaystyle {\phi}(a) \equiv {\phi}(a^{(k)} + a_k p^k) \mod{p^{k+1}}$](img124.png) |
(14) |
Since
![$ {\phi}(a) = 0$](img118.png)
this shows that
![$ {\phi}(a^{(k)} + a_k p^k)$](img125.png)
is in the ideal generated by
![$ p^{k+1}$](img126.png)
.
Similarly,
![$ {\phi}(a^{(k)})$](img127.png)
is in the ideal generated by
![$ p^k$](img8.png)
.
Therefore we can divide
![$ {\phi}(a^{(k)} + a_k p^k)$](img125.png)
and
![$ {\phi}(a^{(k)})$](img127.png)
by
![$ p^k$](img8.png)
, leading to
![$\displaystyle \frac{{\phi}(a^{(k)} + a_k p^k)}{p^k} = \frac{{\phi}(a^{(k)})}{p^k} + {\phi}'(a^{(k)}) a_k + {\psi}(a^{(k)} + a_k p^k) (a_k )^2 p^k.$](img128.png) |
(15) |
Now observe that
![$\displaystyle \frac{{\phi}(a^{(k)} + a_k p^k)}{p^k} \equiv 0 \equiv {\psi}(a^{(k)} + a_k p^k) (a_k )^2 p^k \mod{p}.$](img129.png) |
(16) |
Let us denote by
![$ {\Phi}_p$](img130.png)
the canonical homomorphism from
![$ R$](img1.png)
to
![$ R/\langle p \rangle$](img131.png)
.
Then we obtain
![$\displaystyle {\Phi}_p(\frac{{\phi}(a^{(k)})}{p^k}) = - {\Phi}_p({\phi}'(a^{(k)})) {\Phi}_p(a_k)$](img132.png) |
(17) |
Now, since
![$ a \equiv a_0 \mod{p}$](img119.png)
holds we have
![$\displaystyle {\Phi}_p({\phi}'(a^{(k)})) \equiv {\Phi}_p({\phi}'(a_0)) \mod{p}$](img133.png) |
(18) |
Since
![$ {\phi}'(a_0) \neq 0 \mod{p}$](img120.png)
holds we can solve
Equatiion
17
for
![$ {\Phi}_p(a_k)$](img134.png)
.
Finally, from Theorem
1,
we have
![$ {\deg}(a_k,p) = 0$](img135.png)
such that we can
view
![$ {\Phi}_p(a_k)$](img134.png)
as
![$ a_k$](img136.png)
. This is straightforward if
![$ R = {\mbox{${\mathbb{Z}}$}}$](img137.png)
or if
![$ R = {\bf k}[x]$](img138.png)
where
![$ {\bf k}$](img62.png)
is a field, since we can choose for
![$ {\Phi}_p(x)$](img139.png)
the remainder of
![$ x$](img140.png)
modulo
![$ p$](img5.png)
.
Proof.
We proceed by induction on
![$ {\ell} \geq 1$](img157.png)
.
For
![$ {\ell} = 1$](img158.png)
the claim follows from the hypothesis of the theorem.
So let
![$ {\ell} \geq 1$](img157.png)
be such that the claim is true.
Hence there exist
![$ a^{({\ell})}_1, \ldots, a^{({\ell})}_r \in R$](img152.png)
such that
![$\displaystyle f_i(a^{({\ell})}_1, \ldots, a^{({\ell})}_r) \equiv 0 \mod {{\cal I}^{\ell}}, \ \ i = 1, \ldots, n$](img159.png) |
(20) |
and
![$\displaystyle a^{({\ell})}_j \equiv a_j \mod {{\cal I}}, \ \ i = 1, \ldots, r$](img160.png) |
(21) |
Since
![$ {\cal I}$](img104.png)
is finitely generated, then so is
![$ {\cal I}^{\ell}$](img161.png)
and let
![$ g_1, \ldots, g_s \in R$](img162.png)
such that
![$\displaystyle {\cal I}^{\ell} = \langle g_1, \ldots, g_s \rangle$](img163.png) |
(22) |
Therefore, for every
![$ i = 1, \ldots, r$](img164.png)
, there exist
![$ q_{i1}, \ldots q_{is} \in R$](img165.png)
such that
![$\displaystyle f_i(a^{({\ell})}_1, \ldots, a^{({\ell})}_r) \ = \ {\Sigma}_{k=1}^{k=s} q_{ik} g_k$](img166.png) |
(23) |
For each
![$ j = 1, \ldots, r$](img155.png)
we want to compute
![$ B_j \in R$](img167.png)
such that
![$\displaystyle a^{({\ell}+1)}_j = a^{({\ell})}_j + B_j$](img168.png) |
(24) |
is the desired
next approximation.
We impose
![$ B_j \in {\cal I}^{\ell}$](img169.png)
so let
![$ b_{j1}, \ldots, b_{js} \in R$](img170.png)
be such that
![$\displaystyle B_j \ = \ {\Sigma}_{k=1}^{k=s} b_{jk} g_k$](img171.png) |
(25) |
Using Proposition
5
we obtain
![\begin{displaymath}\begin{array}{rcl} f_i(a^{({\ell}+1)}_1, \ldots, a^{({\ell}+1...
... b_{jk} \right) g_k \mod{{ {\cal I}^{{\ell}+1}}} \\ \end{array}\end{displaymath}](img172.png) |
(26) |
where
![$ \left( u^{({\ell})}_{ij} \right)$](img173.png)
is the Jacobian matrix
of
![$ (f_1, \ldots, f_n)$](img174.png)
at
![$ (a^{({\ell})}_1, \ldots, a^{({\ell})}_r)$](img175.png)
.
Hence, solving for
such that
leads to solving the system of linear equations:
![$\displaystyle q_{ik} + {\Sigma}_{j=1}^{j=r} u^{({\ell})}_{ij} b_{jk} \equiv 0 \mod{{ {\cal I} }}$](img178.png) |
(27) |
for
![$ k = 1, \ldots, s$](img179.png)
and
![$ i = 1, \ldots, n$](img153.png)
.
Now using
for
we obtain
![$\displaystyle u^{({\ell})}_{ij} \equiv u_{ij} \mod{{\cal I}}$](img180.png) |
(28) |
Therefore the linear system equations
given by Relation (
27)
has solutions.