Finding lambda for the validation of Hensel

Latex/pseudocode.tex

@@ -402,6 +402,140 @@ Cherché: $\boldsymbol{g}, \boldsymbol{h} \in \mathbb{IK}[x,y]/(x^{k+1})$ tel qu
 12. end for
 
 
+\subsection*{Finding Lambda}
+
+Nous voulons trouver $\Lambda$ tel que: \begin{equation*}
+      |N_P(x)-N_P(x')|\leq \Lambda(r) |x-x'|.
+\end{equation*}
+Nous avons $x = \left(\begin{matrix}
+      g\\
+      h
+\end{matrix}\right)$
+et $P(g,h) = gh-f$ et en conséquent
+$N_P(g,h) = \left(\begin{matrix}
+      g\\
+      h
+\end{matrix}\right)-
+\left(\begin{matrix}
+      t(gh-f)\%\tilde{g}\\
+      s(gh-f)\%\tilde{h}
+\end{matrix}\right)$.\\
+Finalement, on arrive à
+\begin{equation*}
+      |N_P(g,h) - N_P(g',h')| = \left|\left(\begin{matrix}
+            g-g'\\
+            h-h'
+      \end{matrix}\right)
+      +
+      \left(\begin{matrix}
+            t(-gh+g'h')\%\tilde{g}\\
+            s(-gh+g'h')\%\tilde{h}
+      \end{matrix}\right)\right|.
+\end{equation*} 
+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}
+            g\\
+            h
+      \end{matrix}\right)+ \left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) - \left(\begin{matrix}
+            t\left(\left(g+\delta_g\right)\left(h+\delta_h\right)-f\right)\%\tilde{g}\\
+            s\left(\left(g+\delta_g\right)\left(h+\delta_h\right)-f\right)\%\tilde{h}
+      \end{matrix}\right)\\
+      &=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).\\
+\end{align*}
+Regardons les parties de \begin{equation*}
+      \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): 
+\end{equation*}
+1.
+\begin{align*}
+      th\delta_g\%\tilde{g} &= t\left(\tilde{h}\delta_g+\left(h-\tilde{h}\right)\delta_g\right)\%\tilde{g} = t\tilde{h}\delta_g\%\tilde{g} + t(h-\tilde{h})\delta_g\%\tilde{g}\\
+      &\overset{t\tilde{h} = 1+\epsilon - s\tilde{g}}{=} (1+\epsilon-s\tilde{g})\delta_g\%\tilde{g} +t(h-\tilde{h})\delta_g\%\tilde{g}\\
+      &\overset{s\tilde{g}\delta_g\%\tilde{g}=0}{=} (1+\epsilon)\delta_g\%\tilde{g} +t(h-\tilde{h})\delta_g\%\tilde{g}\\
+      &=\delta_g\%\tilde{g} + \epsilon\delta_g\%\tilde{g}  +t(h-\tilde{h})\delta_g\%\tilde{g}
+\end{align*}
+2.
+\begin{align*}
+      tg\delta_h\%\tilde{g} = t\left(\tilde{g}\delta_h +\left(g-\tilde{g}\right)\delta_h\right)\%\tilde{g} \overset{\tilde{g}\%\tilde{g}=0}{=} t\left(g-\tilde{g}\right)\delta_h\%\tilde{g}
+\end{align*}
+3.
+\begin{align*}
+      t\delta_g\delta_h\%\tilde{g}
+\end{align*}
+4.
+\begin{align*}
+      sh\delta_g\%\tilde{h} &= s\left(\tilde{h} + \left(h-\tilde{h}\right)\right)\delta_g\%\tilde{h} \overset{\tilde{h}\%\tilde{h}= 0}{=} s\left(h-\tilde{h}\right)\delta_g\%\tilde{h} 
+\end{align*}
+5.
+\begin{align*}
+      sg\delta_h\%\tilde{h} &= s\left(\tilde{g}+\left(g-\tilde{g}\right)\right)\delta_h\%\tilde{h}\\
+      &= s\tilde{g}\delta_h\%\tilde{h} + \left(g-\tilde{g}\right)\delta_h\%\tilde{h}\\
+      &\overset{s\tilde{g} = 1+\epsilon-t\tilde{h}}{=} \left(1+\epsilon-t\tilde{h}\right)\delta_h\%\tilde{h} +  s\left(g-\tilde{g}\right)\delta_h\%\tilde{h}\\
+      &\overset{t\tilde{h}\%\tilde{h}=0}{=} \left(1+\epsilon\right)\delta_h\%\tilde{h} +  s\left(g-\tilde{g}\right)\delta_h\%\tilde{h}
+\end{align*}
+6.
+\begin{align*}
+      s\delta_g\delta_h\%\tilde{h}
+\end{align*}
+Comme $\left|\delta_g\right| < r_g$, $\left|\delta_h\right| < r_h$, $\left|g-\tilde{g}\right| < r_g$, et $\left|h-\tilde{h}\right| < r_h$,
+nous avons:
+\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|\\
+      & \leq \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)\\
+      &=  \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)
+\end{align*}
+
+
+
+
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