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Commit 32bcaeeb authored by Krueger Jasmin's avatar Krueger Jasmin
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applied changes to the finding process of lamda for the validation of Hensel

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......@@ -435,7 +435,7 @@ Finalement, on arrive à
On sait que $\tilde{s}\tilde{g}+\tilde{h}\tilde{t} = 1$ et on a une approximation $s \approx \tilde{s}, t \approx \tilde{t}$ avec $s\tilde{g} + \tilde{h}t = 1 + \epsilon$.\\
Alors, nous regardons:
\begin{align*}
N_P(g+\delta_g,h+\delta_h) &=\left(\begin{matrix}
N_P(g',h') = N_P(g+\delta_g,h+\delta_h) &=\left(\begin{matrix}
g\\
h
\end{matrix}\right)+ \left(\begin{matrix}
......@@ -514,43 +514,77 @@ nous avons:
\left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
& \leq \left|\left(\begin{matrix}
r_g\\
r_h
\end{matrix}\right)
+
\left(\begin{matrix}
\left(r_g + \epsilon r_g +tr_hr_g+tr_gr_h+t r_g r_h\right)\%\tilde{g}\\
\left(s r_h r_g+\left(1+\epsilon\right)r_h + s r_g r_h+ s r_g r_h\right)\%\tilde{h}
&\overset{\delta_g\%\tilde{g}=\delta_g}{\overset{\delta_h\%\tilde{h}= \delta_h}{=}} \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) + \left(\begin{matrix}
\left(\delta_g + \epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\left(s\left(h-\tilde{h}\right)\delta_g+\left(1+\epsilon\right)\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&= \left| \left(\begin{matrix}
r_g\\
r_h
\end{matrix}\right)
+ \left(\begin{matrix}
\left((1+\epsilon)r_g + 3tr_gr_h\right)\%\tilde{g}\\
\left((1+\epsilon)r_h + 3sr_gr_h\right)\%\tilde{h}
&\overset{\delta_g\%\tilde{g}= \delta_g}{\overset{\delta_h\%\tilde{h}=\delta_h}{=}} \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right) + \left(\begin{matrix}
\delta_g +\left(\epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\delta_h + \left(s\left(h-\tilde{h}\right)\delta_g+\epsilon\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&= \left|\left(\begin{matrix}
r_g\\
r_h
\end{matrix}\right)
+ \left(\begin{matrix}
\left((1+\epsilon)r_g + 3tr_gr_h\right)-\tilde{g}\cdot rev\left(rev\left((1+\epsilon)r_g + 3tr_gr_h\right)\cdot \dot{g}\right)\\
\left((1+\epsilon)r_h + 3sr_gr_h\right) -\tilde{h}\cdot rev\left(rev\left((1+\epsilon)r_h + 3sr_gr_h\right)\cdot\dot{h}\right)
\left(\epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)\%\tilde{g}\\
\left(s\left(h-\tilde{h}\right)\delta_g+\epsilon\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)\%\tilde{h}
\end{matrix}\right)\right|\\
&= \left|\left(\begin{matrix}
r_g\\
r_h
\end{matrix}\right)
+ \left(\begin{matrix}
(1+\epsilon +3tr_h)r_g-\tilde{g}\cdot rev\left(rev\left((1+\epsilon+3tr_h)r_g\right) \cdot \dot{g}\right)\\
(1+\epsilon+3sr_g)r_h -\tilde{h}\cdot rev\left(rev\left((1+\epsilon+3sr_g)r_h\right)\cdot\dot{h}\right)
\left(\epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right) \\
\left(s\left(h-\tilde{h}\right)\delta_g+\epsilon\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)
\end{matrix}\right.\right.\\
& \qquad \qquad \qquad \qquad \left.\left.\begin{matrix}
- \tilde{g}\cdot\overline{\overline{\left(\epsilon\delta_g +t(h-\tilde{h})\delta_g+t\left(g-\tilde{g}\right)\delta_h+t\delta_g\delta_h\right)}\cdot\dot{g}}\\
-\tilde{h}\cdot \overline{\overline{\left(s\left(h-\tilde{h}\right)\delta_g+\epsilon\delta_h + s\left(g-\tilde{g}\right)\delta_h+s\delta_g\delta_h\right)}\cdot\dot{h}}
\end{matrix}\right)\right|\\
&= \left| \left(\begin{matrix}
(2+\epsilon +3tr_h)r_g-\tilde{g}\cdot rev\left(rev\left((1+\epsilon+3tr_h)r_g\right) \cdot \dot{g}\right)\\
(2+\epsilon+3sr_g)r_h -\tilde{h}\cdot rev\left(rev\left((1+\epsilon+3sr_g)r_h\right)\cdot\dot{h}\right)
\end{matrix}\right)\right|\\
\left(\epsilon + t(h-\tilde{h})+t\delta_h\right)\delta_g + t(g-\tilde{g})\delta_h\\
\left(\epsilon +s(g-\tilde{g}) +s\delta_g\right)\delta_h + s(h-\tilde{h})\delta_g
\end{matrix}\right.\right.\\
& \qquad \qquad \qquad \qquad \left.\left. \begin{matrix}
-\tilde{g}\cdot\overline{\overline{\left(\left(\epsilon + t(h-\tilde{h})+t\delta_h\right)\delta_g + t(g-\tilde{g})\delta_h\right)}\cdot\dot{g}}\\
-\tilde{h}\cdot\overline{\overline{\left(\left(\epsilon +s(g-\tilde{g}) +s\delta_g\right)\delta_h + s(h-\tilde{h})\delta_g\right)}\cdot \dot{h}}
\end{matrix}\right)\right|
% & \leq \left|\left(\begin{matrix}
% r_g\\
% r_h
% \end{matrix}\right)
% +
% \left(\begin{matrix}
% \left(r_g + \epsilon r_g +tr_hr_g+tr_gr_h+t r_g r_h\right)\%\tilde{g}\\
% \left(s r_h r_g+\left(1+\epsilon\right)r_h + s r_g r_h+ s r_g r_h\right)\%\tilde{h}
% \end{matrix}\right)\right|\\
% &= \left| \left(\begin{matrix}
% r_g\\
% r_h
% \end{matrix}\right)
% + \left(\begin{matrix}
% \left((1+\epsilon)r_g + 3tr_gr_h\right)\%\tilde{g}\\
% \left((1+\epsilon)r_h + 3sr_gr_h\right)\%\tilde{h}
% \end{matrix}\right) \right|\\
% &= \left| \left(\begin{matrix}
% r_g\\
% r_h
% \end{matrix}\right)
% + \left(\begin{matrix}
% \left((1+\epsilon)r_g + 3tr_gr_h\right)-\tilde{g}\cdot rev\left(rev\left((1+\epsilon)r_g + 3tr_gr_h\right)\cdot \dot{g}\right)\\
% \left((1+\epsilon)r_h + 3sr_gr_h\right) -\tilde{h}\cdot rev\left(rev\left((1+\epsilon)r_h + 3sr_gr_h\right)\cdot\dot{h}\right)
% \end{matrix}\right) \right|\\
% &= \left| \left(\begin{matrix}
% r_g\\
% r_h
% \end{matrix}\right)
% + \left(\begin{matrix}
% (1+\epsilon +3tr_h)r_g-\tilde{g}\cdot rev\left(rev\left((1+\epsilon+3tr_h)r_g\right) \cdot \dot{g}\right)\\
% (1+\epsilon+3sr_g)r_h -\tilde{h}\cdot rev\left(rev\left((1+\epsilon+3sr_g)r_h\right)\cdot\dot{h}\right)
% \end{matrix}\right) \right| \\
% &= \left| \left(\begin{matrix}
% (2+\epsilon +3tr_h)r_g-\tilde{g}\cdot rev\left(rev\left((1+\epsilon+3tr_h)r_g\right) \cdot \dot{g}\right)\\
% (2+\epsilon+3sr_g)r_h -\tilde{h}\cdot rev\left(rev\left((1+\epsilon+3sr_g)r_h\right)\cdot\dot{h}\right)
% \end{matrix}\right)\right|\\
\end{align*}
ou alors, en notant $\dot{g} := rev(\tilde{g})^{-1}$ et $rev(a):= rev_{\deg a}(a)$:
......@@ -626,11 +660,11 @@ ou alors, en notant $\dot{g} := rev(\tilde{g})^{-1}$ et $rev(a):= rev_{\deg a}(a
\tilde{g} & \\
& \tilde{h}
\end{matrix}\right)
\overline{\overline{M}
\overline{
\left(\begin{matrix}
\dot{g} & \\
& \dot{h}
\end{matrix}\right)}\right|\\
\end{matrix}\right)\overline{M}}\right|\\
&= \left|-\left(\begin{matrix}
\delta_g\\
\delta_h
......@@ -642,12 +676,18 @@ ou alors, en notant $\dot{g} := rev(\tilde{g})^{-1}$ et $rev(a):= rev_{\deg a}(a
\left(\begin{matrix}
t\delta_g\delta_h\\
s\delta_g\delta_h
\end{matrix}\right) -
\end{matrix}\right) \right.\\
& \qquad \qquad \qquad \qquad \qquad \left.-
\left(\begin{matrix}
\tilde{g} & \\
& \tilde{h}
\end{matrix}\right)
\overline{\overline{\left(Z \left(\begin{matrix}
\overline{
\left(\begin{matrix}
\dot{g} & \\
& \dot{h}
\end{matrix}\right)
\overline{\left(Z \left(\begin{matrix}
\delta_g\\
\delta_h
\end{matrix}\right)
......@@ -655,11 +695,7 @@ ou alors, en notant $\dot{g} := rev(\tilde{g})^{-1}$ et $rev(a):= rev_{\deg a}(a
\left(\begin{matrix}
t\delta_g\delta_h\\
s\delta_g\delta_h
\end{matrix}\right)\right)}
\left(\begin{matrix}
\dot{g} & \\
& \dot{h}
\end{matrix}\right)}\right|
\end{matrix}\right)\right)}}\right|
\end{align*}
%\printbibliography
......
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