|
|
Last edited by 青青子衿 2025-7-20 23:35covariant of binary quartics
On the equivalence of binary quartics
J. E. Cremona T.A.Fisher
johncremona.github.io/papers/quartequiv.pdf
mathoverflow.net/questions/324640
mathoverflow.net/questions/30891
5. Arithmetic covariants and a syzygy of Cayley and Hermite
math.ucla.edu/~wdduke/preprints/elliptic.pdf
forum.php?mod=redirect&goto=findpost& … d=6670&pid=52733
\begin{align*}
j&=\frac{I^3}{\Delta}=\frac{I^3}{I^3-27J^2}\\
I&=\begin{vmatrix}
a & \frac{c}{6} \\
\frac{c}{6} & e \\
\end{vmatrix} -4 \begin{vmatrix}
\frac{b}{4} & \frac{c}{6} \\
\frac{c}{6} & \frac{d}{4} \\
\end{vmatrix}=a e-\frac{b d}{4}+\frac{c^2}{12}\\
J&=\begin{vmatrix}
a & \frac{b}{4} & \frac{c}{6} \\
\frac{b}{4} & \frac{c}{6} & \frac{d}{4} \\
\frac{c}{6} & \frac{d}{4} & e \\
\end{vmatrix}=\frac{a c e}{6}-\frac{a d^2}{16}-\frac{b^2 e}{16}+\frac{b c d}{48}-\frac{c^3}{216}\\
\end{align*}
\begin{align*}
I&=-\frac{\partial\,\!}{\partial\,\!\lambda}
\left.\det
\left[\begin{pmatrix}
a & \frac{b}{4} & \frac{c}{6} \\
\frac{b}{4} & \frac{c}{6} & \frac{d}{4} \\
\frac{c}{6} & \frac{d}{4} & e \\
\end{pmatrix}+\lambda\begin{pmatrix}
0 & 0 & 2 \\
0 & -1 & 0 \\
2 & 0 & 0 \\
\end{pmatrix}\right]\right|_{\lambda=0}\\
J&=-
\left.\det
\left[\begin{pmatrix}
a & \frac{b}{4} & \frac{c}{6} \\
\frac{b}{4} & \frac{c}{6} & \frac{d}{4} \\
\frac{c}{6} & \frac{d}{4} & e \\
\end{pmatrix}+\lambda\begin{pmatrix}
0 & 0 & 2 \\
0 & -1 & 0 \\
2 & 0 & 0 \\
\end{pmatrix}\right]\right|_{\lambda=0}\\
\operatorname{Covar}_{[2,4]}&=
48\begin{pmatrix}
y^2 & -xy & x^2 \\
\end{pmatrix}
\left[\operatorname{adj}\begin{pmatrix}
a & \frac{b}{4} & \frac{c}{6} \\
\frac{b}{4} & \frac{c}{6} & \frac{d}{4} \\
\frac{c}{6} & \frac{d}{4} & e \\
\end{pmatrix}\right]\begin{pmatrix}
y^2 \\
-xy \\
x^2 \\
\end{pmatrix}\\
&=48\begin{pmatrix}
y^2 & -xy & x^2 \\
\end{pmatrix}\begin{pmatrix}
\frac{8 c e-3 d^2}{48} & \frac{c d-6 b e}{24} & \frac{9 b d-4 c^2}{144} \\
\frac{c d-6 b e}{24} & \frac{36 a e-c^2}{36} & \frac{b c-6 a d}{24} \\
\frac{9 b d-4 c^2}{144} & \frac{b c-6 a d}{24} & \frac{8 a c-3 b^2}{48} \\
\end{pmatrix}\begin{pmatrix}
y^2 \\
-xy \\
x^2 \\
\end{pmatrix}\\
&=\frac{1}{3}\operatorname{HessDet}(ax^4+bx^3y+cx^2y^2+dxy^3+ey^4)\\
\end{align*}
\begin{align*}
\begin{vmatrix}
3 b^2-8 a c & b c-6 a d & b d-16 a e \\
b c-6 a d & -4 a e-2 b d+c^2 & c d-6 b e \\
b d-16 a e & c d-6 b e & 3 d^2-8 c e \\
\end{vmatrix}=4\operatorname{Discr}[a x^4+b x^3+c x^2+d x+e]
\end{align*}
\begin{align*}
\int_{0}^{1}\frac{6t^{2}-t+5}{\sqrt{t^{6}+4t^{4}-2t^{3}+8t^{2}-4t+5}}{\mathrm{d}t}=2\ln\big(2+\sqrt{3}\>\!\big)
\end{align*}
- 1728*\[ScriptCapitalI]^3/(\[ScriptCapitalI]^3 -
- 27 \[ScriptCapitalJ]^2) /. {\[ScriptCapitalI] -> Det[( {
- {a, c/6},
- {c/6, e}
- } )] - 4*Det[( {
- {b/4, c/6},
- {c/6, d/4}
- } )],
- \[ScriptCapitalJ] -> Det[( {
- {a, b/4, c/6},
- {b/4, c/6, d/4},
- {c/6, d/4, e}
- } )]} /. {a -> 1, b -> \[Alpha], c -> \[Beta], d -> \[Gamma],
- e -> \[Delta]} // Factor
- 1728*(4 A^3)/(4 A^3 + 27 B^2) /. {A -> (16 v)/(3 u^4),
- B -> 1/u^4 (1 - (48 v*w)/(27 u^2) - w^3/(
- 27 u^2))} /. {u -> \[Alpha]^3 - 4*\[Alpha]*\[Beta] +
- 8*\[Gamma], v -> 3*\[Alpha]*\[Gamma] - \[Beta]^2 - 12*\[Delta],
- w -> 3*\[Alpha]^2 - 8*\[Beta]} // Factor
- 1728*(4 A^3)/(4 A^3 + 27 B^2) /. {A -> 3 k^2*j/(1728 - j),
- B -> 2 k^3*j/(1728 - j)} // Factor
|
|
