三次曲线的Rob微分不变量

青青子衿

\begin{align*}
\operatorname{Rob}(y)=\begin{vmatrix}
3y^{(2)} & 4y^{(3)} & y^{(4)} & 4 (y^2)^{(3)} & (y^2)^{(4)} & (y^3)^{(4)} \\

4y^{(3)} & 5y^{(4)} & y^{(5)} & 5(y^2)^{(4)} & (y^2)^{(5)} & (y^3)^{(5)} \\

5y^{(4)} & 6y^{(5)} & y^{(6)} & 6 (y^2)^{(5)} & (y^2)^{(6)} & (y^3)^{(6)} \\

6y^{(5)} & 7y^{(6)} & y^{(7)} & 7 (y^2)^{(6)} & (y^2)^{(7)} & (y^3)^{(7)} \\

7y^{(6)} & 8y^{(7)} & y^{(8)} & 8 (y^2)^{(7)} & (y^2)^{(8)} & (y^3)^{(8)} \\

8y^{(7)} & 9y^{(8)} & y^{(9)} & 9 (y^2)^{(8)} & (y^2)^{(9)} & (y^3)^{(9)} \\
\end{vmatrix}
\end{align*}

1. Det[( {
2.      {3 D[y[x], {x, 2}], 4 D[y[x], {x, 3}], D[y[x], {x, 4}],
3.       4 D[y[x]^2, {x, 3}], D[y[x]^2, {x, 4}], D[y[x]^3, {x, 4}]},
4.      {4 D[y[x], {x, 3}], 5 D[y[x], {x, 4}], D[y[x], {x, 5}],
5.       5 D[y[x]^2, {x, 4}], D[y[x]^2, {x, 5}], D[y[x]^3, {x, 5}]},
6.      {5 D[y[x], {x, 4}], 6 D[y[x], {x, 5}], D[y[x], {x, 6}],
7.       6 D[y[x]^2, {x, 5}], D[y[x]^2, {x, 6}], D[y[x]^3, {x, 6}]},
8.      {6 D[y[x], {x, 5}], 7 D[y[x], {x, 6}], D[y[x], {x, 7}],
9.       7 D[y[x]^2, {x, 6}], D[y[x]^2, {x, 7}], D[y[x]^3, {x, 7}]},
10.      {7 D[y[x], {x, 6}], 8 D[y[x], {x, 7}], D[y[x], {x, 8}],
11.       8 D[y[x]^2, {x, 7}], D[y[x]^2, {x, 8}], D[y[x]^3, {x, 8}]},
12.      {8 D[y[x], {x, 7}], 9 D[y[x], {x, 8}], D[y[x], {x, 9}],
13.       9 D[y[x]^2, {x, 8}], D[y[x]^2, {x, 9}], D[y[x]^3, {x, 9}]}
14.     } ) ] /. y -> (Power[#^3 + #, 1/3] &) // Factor

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hbghlyj

这等于0？

hbghlyj

如何证明三次曲线的Rob不变量等于零