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Background DiffResolvent.nb, here gives a method that solve quintic equations using differential equations. There’s a progress to transform polynomial equation into differential equation. Then he gave an example: $ $ x^5-5 x^3+5 x-t = 0$ $ The roots of the equation satisfy the differential equation: $ $ 25 \left(4-t^2\right) x”(t)-25 t x'(t)+x(t)=0$ $ Solve thatRead more

I am looking for explicit formulas for the four basic invariants $ I_4, I_8, I_{12}, I_{18}$ of a generic binary quintic form, either given in the shape $ $ \displaystyle F(x,y) = ax^5 + 5bx^4y + 10cx^3y^2 + 10dx^2y^3 + 5exy^4 + fy^5$ $ or $ $ \displaystyle F(x,y) = ax^5 + bx^4y + cx^3y^2Read more