本帖最后由 青青子衿 于 2019-11-4 10:41 编辑
回复 1# 青青子衿
幂简洁形式只是相对的,其实也没有多简洁……
\begin{align*}
&\left(3g_0g_1g_2h_3-3g_0{g_2}^2h_2-4{g_1}^3h_3+6{g_1}^2g_2h_2-3g_1{g_2}^2h_1+{g_2}^3h_0\right)^2\\
=\,&\left({g_1}^2-g_0g_2\right)\left(g_0g_2h_3-4{g_1}^2h_3+6g_1g_2h_2-3{g_2}^2h_1\right)^2
\end{align*}
\begin{align*}
\left(\dfrac{3g_0g_1g_2h_3-3g_0{g_2}^2h_2-4{g_1}^3h_3+6{g_1}^2g_2h_2-3g_1{g_2}^2h_1+{g_2}^3h_0}{g_0g_2h_3-4{g_1}^2h_3+6g_1g_2h_2-3{g_2}^2h_1}\right)^2={g_1}^2-g_0g_2\geqslant0
\end{align*}
...
\begin{gather*}
\begin{vmatrix}
f_{xx}&2f_{xy}&f_{yy}&&\\
&f_{xx}&2f_{xy}&f_{yy}&\\
&&f_{xx}&2f_{xy}&f_{yy}\\
f_{xxx}&3f_{xxy}&3f_{xyy}&f_{yyy}&\\
&f_{xxx}&3f_{xxy}&3f_{xyy}&f_{yyy}\\
\end{vmatrix}\\
\\
=\dfrac{1}{{f_{yy}}^3}\left(\begin{split}
3f_{xx}f_{xy}f_{yy}\cdot\,\!f_{yyy}-3f_{xx}{f_{yy}}^2\cdot\,\!f_{xyy}-4{f_{xy}}^3\cdot\,\!f_{yyy}
+6{f_{xy}}^2f_{yy}\cdot\,\!f_{xyy}-3f_{xy}{f_{yy}}^2\cdot\,\!f_{xxy}+{f_{yy}}^3\cdot\,\!f_{xxx}
\end{split}\right)^2\\
-\dfrac{{f_{xy}}^2-f_{xx}f_{yy}}{{f_{yy}}^3}\left(f_{xx}f_{yy}\cdot\,\!f_{yyy}-4{f_{xy}}^2\cdot\,\!f_{yyy}+6f_{xy}f_{yy}\cdot\,\!f_{xyy}-3{f_{yy}}^2\cdot\,\!f_{xxy}\right)^2
\end{gather*}
\begin{gather*}
\operatorname{Reslt}\left(\left.
\begin{matrix}
f_{xx}+2f_{xy}p+f_{yy}p^2\\
f_{xxx}+3f_{xxy}p+3f_{xyy}p^2+f_{yyy}p^3\\
\end{matrix}\right|\,p\right)
=
\begin{vmatrix}
f_{xx}&2f_{xy}&f_{yy}&&\\
&f_{xx}&2f_{xy}&f_{yy}&\\
&&f_{xx}&2f_{xy}&f_{yy}\\
f_{xxx}&3f_{xxy}&3f_{xyy}&f_{yyy}&\\
&f_{xxx}&3f_{xxy}&3f_{xyy}&f_{yyy}\\
\end{vmatrix}\equiv0\\
\\
\Downarrow\\
\left(\begin{split}
3f_{xx}f_{xy}f_{yy}\cdot\,\!f_{yyy}-3f_{xx}{f_{yy}}^2\cdot\,\!f_{xyy}-4{f_{xy}}^3\cdot\,\!f_{yyy}
+6{f_{xy}}^2f_{yy}\cdot\,\!f_{xyy}-3f_{xy}{f_{yy}}^2\cdot\,\!f_{xxy}+{f_{yy}}^3\cdot\,\!f_{xxx}
\end{split}\right)^2\\
\equiv\left({f_{xy}}^2-f_{xx}f_{yy}\right)\left(f_{xx}f_{yy}\cdot\,\!f_{yyy}-4{f_{xy}}^2\cdot\,\!f_{yyy}+6f_{xy}f_{yy}\cdot\,\!f_{xyy}-3{f_{yy}}^2\cdot\,\!f_{xxy}\right)^2
\end{gather*} |