Further work on finding lambda for validation of Hensel

Latex/pseudocode.tex

@@ -533,8 +533,114 @@ nous avons:
       \end{matrix}\right)
 \end{align*}
 
-
-
+ou alors, en notant $\dot{g} := rev(\tilde{g})^{-1}$ et $rev(a):= rev_{\deg a}(a)$:
+\begin{align*}
+      \left|N_P(g,h)-N_P(g',h')\right| &= \left|N_P(g,h)-N_P(g+\delta_g,h+\delta_h)\right|\\
+      &= \left|N_P(g,h) - \left(N_P(g,h) + \left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) - \left(\begin{matrix}
+            t\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{g}\\
+            s\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{h}
+      \end{matrix}\right)\right)\right|\\
+      &= \left|-\left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) + \left(\begin{matrix}
+            t\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{g}\\
+            s\left(h\delta_g+g\delta_h+\delta_g\delta_h\right)\%\tilde{h}
+      \end{matrix}\right)\right|\\
+      &= \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(
+      \Bigg[\left(\begin{matrix}
+            M_{1\_}\\
+            M_{2\_}
+      \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)\\
+            \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)
+      \end{matrix}\right)\Bigg]\\
+      &\approx \left|-\left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) + 
+      \left(\begin{matrix}
+            M_{1\_} - \tilde{g}\cdot rev\left(rev\left(M_{1\_}\right)\cdot\dot{g}\right)\\
+            M_{2\_} - \tilde{h}\cdot rev\left(rev\left(M_{2\_}\right)\cdot\dot{h}\right)
+      \end{matrix}\right)\right|\\
+      \Bigg[M &= \left(\begin{matrix}
+            1+\epsilon+t(h-\tilde{h}) & t(g-\tilde{g})\\
+            s(h-\tilde{h}) & 1+\epsilon+s(g-\tilde{g})
+      \end{matrix}\right)
+      \left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) 
+      +
+      \left(\begin{matrix}
+            t\delta_g\delta_h\\
+            s\delta_g\delta_h
+      \end{matrix}\right)\\
+      & =: Z \left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) 
+      +
+      \left(\begin{matrix}
+            t\delta_g\delta_h\\
+            s\delta_g\delta_h
+      \end{matrix}\right)\\
+      \overline{a} &:= rev(a)\Bigg]\\
+      &= \left|-\left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) + M -
+      \left(\begin{matrix}
+            \tilde{g} & \\
+             & \tilde{h}
+      \end{matrix}\right)
+      \overline{\overline{M}
+      \left(\begin{matrix}
+            \dot{g} & \\
+             & \dot{h}
+      \end{matrix}\right)}\right|\\
+      &= \left|-\left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) + Z \left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) 
+      +
+      \left(\begin{matrix}
+            t\delta_g\delta_h\\
+            s\delta_g\delta_h
+      \end{matrix}\right) -
+      \left(\begin{matrix}
+            \tilde{g} & \\
+             & \tilde{h}
+      \end{matrix}\right)
+      \overline{\overline{\left(Z \left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) 
+      +
+      \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{align*}
 
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