pseudocode Hensel validation

Latex/pseudocode.tex

@@ -708,6 +708,78 @@ Comme $\left|\delta_g\right| < r_g$, $\left|\delta_h\right| < r_h$, $\left|g-\ti
 % \end{align*}
 
 
+Pour calculer $\Lambda$ explicitement, nous savons
+\begin{align*}
+      |N_P(g,h)-N_P(g',h')| &\leq \sup_{\bar{g},\bar{h}\in \mathcal{B}(\tilde{g},\tilde{h},r)}|\nabla N_P(\bar{g},\bar{h})|\left|\begin{matrix}
+            g-g'\\
+            h-h'
+      \end{matrix}\right|
+\end{align*}
+
+
+\begin{equation*}
+      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)
+\end{equation*}
+
+
+\begin{align*}
+      \nabla N_P(g,h)\left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) &= 
+      \left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) - \left(\begin{matrix}
+            t(h\delta_g +g\delta_h)\%\tilde{g}\\
+            s(h\delta_g + g\delta_h)\%\tilde{h}
+      \end{matrix}\right)\\
+      &= \left|\left(\begin{matrix}
+            (\epsilon\delta_g+t(h-\tilde{h})\delta_g + t(g+\tilde{g})\delta_h)\%\tilde{g}\\
+            (\epsilon\delta_h +s(g-\tilde{g})\delta_h + s(h-\tilde{h})\delta_g)\%\tilde{h}
+      \end{matrix}\right)\right|\\
+      \Rightarrow \left|\nabla N_P(g,h)\left(\begin{matrix}
+            \delta_g\\
+            \delta_h
+      \end{matrix}\right) \right|
+      &\leq \left| \begin{matrix}
+            (\epsilon\delta_g + t([\pm r_h] \delta_g +[\pm r_g]\delta_g))\%\tilde{g}\\
+            (\epsilon\delta_h + s([\pm r_h]\delta_g +[\pm r_g]\delta_h))\%\tilde{h}
+      \end{matrix}\right|\\
+      &\leq \left| \begin{matrix}
+            (\epsilon[\pm r_g] + 2t[\pm r_h][\pm r_g])\%\tilde{g}\\
+            (\epsilon[\pm r_h] + 2s[\pm r_h][\pm r_g])\%\tilde{h}
+      \end{matrix}\right|\\
+      &= \Lambda(r)
+\end{align*}
+
+\subsection*{Pseudocode}
+$L = \mathbb{K}[x]/(x^k)$\\
+$IL = \mathbb{IK}[x]/(x^k)$\\
+Problème : $f\equiv gh \mod x^k$\\
+Connu : $f \in L[y]_{\leq m+n}$, et des approximations numériques $\tilde{g}\in L[y]_{\leq n}, \tilde{h}\in L[y]_{\leq m}$, $h$ unitaire, de $g$ et $h$.\\
+Cherché : $\boldsymbol{g} \in IL[y]_{\leq n}, \boldsymbol{h}\in IL[y]_{\leq m}$ tel que $\tilde{g}_{i,j},g_{i,j}^* \in \boldsymbol{g}_{i,j}$ et $\tilde{h}_{i,j},h_{i,j}^* \in \boldsymbol{h}_{i,j}$\\
+Nous avons $\delta = \left|\left( \begin{matrix}
+      t(\tilde{g}\tilde{h}-f)\%\tilde{g}\\
+      s(\tilde{g}\tilde{h}-f)\%\tilde{h}
+\end{matrix}\right)\right|$ et $\mu(r) = \left|\left(\begin{matrix}
+      \left(\epsilon[\pm r_g]  +3t[\pm r_h][\pm r_g]\right)\%\tilde{g}\\
+      \left(3s[\pm r_h][\pm r_g]+\epsilon[\pm r_h] \right)\%\tilde{h}
+\end{matrix}\right)\right|$\\
+Input: \\
+Output:\\
+1. $ q, r = fastdivrem(se,h)$\\
+2. $\delta_h = r$\\
+3. $\delta_g = te + qg$\\
+4. $\mu = $
+
 
 
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